i : '4 ' iiiiiiiiiii ^Xi n 27 13. Number of linearly independent seminvariants ... . 28 14-15. Derivation of modular invariants from semin- variants 28 LECTURE III Inyaeiants of a Modular Group. Formal Invariants and COVARL^NTS of JNIoDULAR FoRMS. APPLICATIONS 1-4. Invariants of certain modular groups; problem of Hurwitz 33 5-11. Formal invariants and seminvariants of binary modular forms 40 12. Theorem of Miss Sanderson 54 13. Fundamental systems of modular CO variants 55 14. Form problem for the total binary modular group 58 15. Invariantive classification of forms 61 LECTURE IV Modular Geometry and Covariantive Theory of a Quad- ratic Form in m Variables INIodulo 2 1-2. Introduction. The polar locus 65 3. Odd number of variables; apex; linear tangential equation 66 4. Covariant line of a conic 69 5. Even number of variables 70 6. Covariant plane of a degenerate quadric surface . . 71 7. A configuration defined by the quinary surface .... 72 8. Certain formal and modular covariants of a conic 73 9-32. Fundamental system of covariants of a conic 76 33. References on modular geometry 98 CONTENTS. Ill LECTURE V A Theory of Plane Cubic Curves with a Real Inflexion Point Valid in Ordinary and in Modular Geometry 1. Normal form of a ternary cubic 99 2. The invariants s and t 99 3- The four inflexion triangles 100 4. The parameter 5 in the normal form 101 5-9. Criteria for 9, 3 or 1 real inflexion points; sub-cases 101 ON INVARIANTS AND THE THEORY OF NUMBERS BY LEONARD EUGENE DICKSON INTRODUCTION A simple theory of invariants for the modular forms and linear transformations employed in the theory of numbers should be of an importance commensurate with that of the theory of invariants in modern algebra and analytic projective geometry, and should have the advantage of introducing into the theory of numbers methods uniform with those of algebra and geometry. In considering the invariants of a modular form (a homo- geneous polynomial with integral coefficients taken modulo p, where p is a prime), we see at once that the rational integral invariants of the corresponding algebraic form with arbitrary variables as coefficients give rise to as many modular invariants of the modular form, and that there are numerous additional invariants peculiar to the case of the theory of numbers. More- over, nearly all of the processes of the theory of algebraic in- variants, whether symbolic or not, either fail for modular in- variants or else become so complicated as to be useless. For instance, the annihilators are no longer linear differential oper- ators. The attempt to construct a simple theory of modular invariants from the standpoints in vogue in the algebraic theory was a failure, although useful special results were obtained in this laborious way. Later I discovered a new standpoint which led to a remarkably simple theory of modular invariants. This standpoint is of function-theoretic character, employing the 2 1 2 THE MADISON COLLOQUIUM. values of the invariant, and using linear transformations only in the preliminary problem of separating into classes the particular forms obtained by assigning special values to the coefficients of the ground form. As to the practical value of the new theory as a working tool, it may be observed that the problem to find a fundamental system of modular seminvariants of a binary form is from the new standpoint a much simpler problem than the cor- responding one in the algebraic case; indeed, we shall exhibit explicitly the fundamental system of modular seminvariants for a binary form of general degree. It will now be clear why these Lectures make no use of the technical theories of algebraic invariants. On the contrary, they afford an introduction to that subject from a new standpoint and, in particular, throw considerable new light on the relations between the subjects of rational integral invariants and tran- scendental invariants of algebraic forms and the corresponding questions for seminvariants. Again, I shall make no use of technical theory of numbers, presupposing merely the concepts of congruence and primitive roots, Fermat's theorem, and (in Lectures III and V) the concept of quadratic residues. The developments given in these Lectures are new, with exceptions in the case of Lecture I, which presents an intro- duction to the theory, and in the case of the earlier and final sections of Lecture III. But in these cases the exposition is considerably simpler and more elementary than that in my published papers on the same topics. The contacts with the work of other writers will be indicated at the appropriate places. Much light is thrown upon the unsolved problem of Hurwitz concerning formal invariants. In many parts of these Lectures, I have not aimed at complete generality and exhaustiveness, but rather at an illumination of typical and suggestive topics, treated by that particular method which I have found to be the best of various possible methods. Surely in a new subject in which most of the possible methods are very complex, it is desirable to put on record an account of the INVARIANTS AND NUMBER THEORY. 3 simple successful methods. Finally, it may be remarked that the present theory is equally simple when the coefficients of the forms and linear transformations are not integers, but are ele- ments of any finite field. I am much indebted to Dr. Sanderson and Professors Cole and Glenn for reading the proof sheets. LECTURE I A THEORY OF INVARIANTS APPLICABLE TO ALGEBRAIC AND MODULAR FORMS Introduction to the Algebraic Side of the Theory by Means of the Example of an Algebraic Quadratic Form in m Variables, §§ 1-3 1. Classes of Algebraic Quadratic Forvis. — Let the coefficients of m (1) gm = 2 ^ijXiXj (J^ji = fiij) be ordinary real or complex numbers. Let the determinant (2) D= \l3ij\ (i,j= 1, '",m) of a particular form qm be of rank r (r > 0); then every minor of order exceeding r is zero, while at least one minor of order r is not zero. There exists a linear transformation of determinant unity which replaces this q^ by a form* (3) ai^Ti^ H h ar.Tr- (ai + 0, . • • , a. + 0). Indeed, if fin 4= 0, we obtain a form lacking .Ti.ro, • • • , XiXm by substituting xi — fin~KfinX2 + • • • + fiiviXm) for .Ti. If fin = 0, fin =r 0, we substitute Xi for Xi and — .ri for .Ti; while, if every fikk = 0, and fin 4= 0, we substitute X2 + Xi for X2; in either case we obtain a form in which the co- eflficient of x{- is not zero. We now have ai.Ti^ + (f>, where ai 4= and involves only X2, • • • , Xm. Proceeding similarly with 0, we ultimately obtain a form (3). Now (3) is replaced by a similar form having ai = 1 by the * Note for later use that each ak and each coefficient of the transformation is a rational function of the /3's with integral coefficients. 4 INVARIANTS AND NUMBER THEORY. 5 transformation .Ti = arhi, .r„, = diKi'm', Xi = x/ (i = 2, • • •, 7?i — 1) of determinant unity. Hence there exists a linear transforma- tion with complex coefficients of determinant unity which replaces qm by (4) .Ti^ + . • • + xl-i + DxJ, X,' + • • • + Xr', according as r = m or r < m. In the first case, the final co- efficient is D since the determinant (2) of a form qm equals that of the form derived from qm by any linear transformation of de- terminant unity. Hence all quadratic forms (1) may be separated into the classes (5) C^.B, Cr (Z) 4=0w = 0, 1, •••,m-l), where, for a particular number Z) 4= 0, the class C^, i, is composed of all forms qm of determinant D, each being transformable into (4i) ; while, for < r < m, the class Cr is composed of all forms of rank r, each being transformable into (42); and, finally, the class Co is composed of the single form with every coefficient zero. In the last case, the determinant D is said to be of rank zero. Using also the fact that the rank of the determinant of a quadratic form is not altered by linear transformation, we con- clude that two quadratic fonns are tramformahle into each other by linear transformations of determinant unity if and only if they belong to the same class (5). 2. Single-valued Invariants of q^. — Using the term function in Dirichlet's sense of correspondence, we shall say that a single- valued function of the undetermined coefficients /3,7 of the general quadratic form q^, is an invariant of qm if has the same value for all sets /3^, jS^^., • • • of coefficients of forms q',„ ql, • • • belonging to the same class.* The values i'„,, o, iv of (f> for the various classes (5) are in general different. For example, the determinant D is an invariant; likewise the single- valued func- * Briefly, if has the same value for all forms in any class. 6 THE MADISON COLLOQUIUM. tion r of the undetermined coefficients ^a which specifies the rank of liSiil- Each consistent set of values of D and r uniquely determines a class (5) and, by definition, each class uniquely determines a value of 0. Hence is a single valued function of D and r. Every single-valued invariant of a system of forms is a single- valued function of the invariants {D and r in our example) which completely characterize the classes. 3. Rational Integral Invariants of qm. — If the invariant <^ is a rational integral function of the coefficients fin, it equals a rational integral function of D. For, if the |S's have any values such that i) =t= 0, has the same value for the form (1) as for the particular form (4i) of the same class. Hence = P{D), where P(-D) is a polynomial in D with numerical coefficients. Since this equation holds for all sets of jS's whose determinant is not zero, it is an identity. Introduction to the Number Theory Side of the Theory of Invariants by Means of the Example OF A Modular Quadratic Form, §§ 4-7 4. Classes of Modular Quadratic Forms qm. — Let Xi, • • • , Xm be indeterminates in the sense of Kronecker. Let each fiij be an integer taken modulo p, where p is an odd prime. Then the expression (1) is called a modular quadratic form. By § 1, there exists a linear transformation, whose coefficients are integers* taken modulo p and whose determinant is congruent to unity, which replaces qm by a quadratic form (3) in which each ak is an integer not divisible by p. Thusf each ak is congruent to a power of a primitive root p of p. After applying a linear trans- formation of determinant unity which permutes Xi^, • • • , Xr^, we may assume that ai, • • •, «« are even powers of p and that aa+i, • • • , ar are odd powers of p. The transformation which * Perhaps initially of the form a/b, where a and b are integers, b not divisible by p. But there exists an integral solution q oi qb = a (mod p). t For p = 5, p = 2, 1 s 2", 2 = 2^, 3 = 2», 4 = 2^ (mod 5). INVARIANTS AND NUMBER THEORY. 7 multiplies a particular Xi {i < m) by p^ and Xm by p~^ is of de- terminant unity. First, let r < m. Applying transformations of the last type to (3), we obtain (6) a:i2 + . . . + xs^ + pxl^, + • • • + pxr\ Under the transformation of determinant unity Xi = aXi + PXj, .Ty = - pXi + aXj, Xm = («2 + ^T'Xm, X? + x^ becomes {o? + ^"){X.^ + ^/). Choose* integers a, /3 so that (7) p(a2 + /32) = 1 (mod 2?). Hence the sum of two terms of (6) with the coefficient p can be transformed into a sum of two squares. Thus by means of a linear transformation, with integral coefficients of determinant unity, Qm can be reduced to one of the forms (8) .Ti2 + . . . + xU, + xr\ xi' + • • • +xU+pXr' (0 composed of all modular quadratic forms whose determinant is a given integer D not divisible by p, each being transformable into (4i), where Cr, i and Cr, -i are composed of all forms transformable into (81) and (82) respectively, and Co is composed of the form all of whose coefficients are zero. Two modular quadratic forms are transformable into each other by linear transformations ivith integral coefficients of determinant unity modulo p if and only if they belong to the same class (9). Indeed, since D and r are invariants,* it remains only to show that the two forms (8) are not transformable into each other. f But if a linear transformation Xi = ^aijXj (i= 1, • • •, m) replaces / = .rr + • • • + xr' by F= X,'+--' + ZLi + pXr', then, for j > r, dXj ~ j=i * dXj dX ~ ^' dX ~ "'' ~ ^ (^ Sr,j> r), r Xi = ^oiijXj (i = 1, • • •, r). Hence under this partial transformation on Xi, • • • , Xr, we would have f = F. Thus the determinant of F would equal \ 0) has a non-vanishing principal minor M of order )•, i. e., one whose diagonal elements lie in the main diagonal of the given determinant.! After an evident linear transformation of determinant unity, we may set (10) M = \^ij\ ^ (mod p) (i,j=l, ..., ,-). In the present problem, /• < m. To g^ apply the transforma- tion Xi = Xi + dXm (i = 1, • • •, r), Xi = Xi (i = r + 1, • • •, /?0 of determinant unity in which the c,- are integers. We get m— 1 '/«— 1 / /• \ 2^ ^ijXiXj + 2 2_/ BjmXjXm -\- [^ BjmCj + Bmm I Xm', i, j=l J=l \ J=i / where r Bjm = S fiijCi + ^jm U = I, •••, m). In view of (10) there are unique values of Ci, • • • , Cr such that Bjm=0 (mod 2>) (i = 1, •• •, /•)• But the determinant of the coefficients of ci, • • •, Cr, 1 in Bim, Bim, • • • , Brm, Bkm (r < k ^ m) ♦Kronecker, Werke, vol. 1, p. 166, p. 357; cf. Gundelfinger, Crelle, vol. 91 (1881), p. 221; Bocher, Introduction to Higher Algebra, p. 58, p. 139. tThe most elementary proof is that by Dickson, Annals of Mathcmalics, ser. 2, vol. 15 (1913), pp. 27, 28. For other short proofs, see Wedderburn, ibid., p. 29, and Kowalewski, Determinantentheorie, pp. 122-124. 10 THE MADISON COLLOQUIUM. is the minor of ^km in the determinant fe-| (hj= 1, "',r,k,m) and hence is zero, being of order r + 1. Hence Bkm — 0. Thus Qm has been transformed into m—l After repetitions of this process, Qm is transformed into* r (11) S fiijXiXj. This form, of determinant M, can be reduced (§ 4) to by a linear transformation on a^i, • • • , Xr with integral coefficients of determinant unity modulo p. Express M as a power p^'^* (e = or 1) of a primitive root. Since r < m, we may replace Xr by p~^Xt and Xm. by p'.T;„ and obtain (8i) or (82) according as € = or € = 1. Now p(p~^)/2 jg j^Q^ congruent to unity, but its square is congruent to unity modulo p, by Fermat's theorem; hence it is = — 1. Thus, in the respective cases, (12) M 2 = + 1 or - 1 (mod p). Hence if a form is of rank r and if M is any chosen r-rowed principal minor not divisible by p, the form is of class Cr, 1 or Cr, -1 according as the first or second alternative (12) holds. 6. Invariantive Criterion for Classes Cr, ±1. — A function which has the value + 1 for any form of class Cr, +1, the value — 1 for any form of class Cr. -1, and the value zero for the remaining classes Cm, d, Co, Ck, ±1 {k ^ r), is an invariant (§ 2). This functionf is * This proof and the results in §§ 4-13 are due to Dickson, Transactions of the American Mathematical Society, vol. 10 (1909), pp. 123-133. t Constructed synthetically in the paper last cited. INVAKIANTS AND NUMBER THEORY. U Ar = {Mi~ + M2~ (1 - il/i^i) + • • • (13) + Mr^a - 1/iM •••(!- m:z\)}u{i - r/'-i), where Mi, • • • , Mn denote the principal minors of order r taken in any sequence, and d ranges over the principal minors of orders exceeding r. For, if any d ^ 0, the rank exceeds r and ^^ = by Fermat's theorem. Next, let every d ^ 0, so that the rank is r or less, and the final factor in (13) is congruent to unity. Then, if every Mi = 0, the rank is less than r and Ar^ 0. But, if Ml ^ 0, Ar = Ml - ^ ± 1 (mod 2>)> by (12), the sign being the same as in Cr, ±i. If Mi = 0, M^ ^ 0, p-i ^, = 3/2 2 = ± 1 (mod jy), etc. Note for later use that (14) Am = D^. 7. Rational Integral Invariants of q,n. — The function (15) 7o = n(l - /3^7') (^, i = 1, ■ . . , m; i ^ j) has the value 1 for the form (of class Co) all of whose coefficients are zero and the value for all remaining forms qm, and hence is an invariant of qm- We now have rational integral invariants (16) D, Ai, •••, Am-u h which completely characterize the classes (9). Hence, by the general theorem in § 12, any rational integral invariant of the modular form qm is a rational integral function of the invariants (16) with integral coefficients. In other words, invariants (16) form a fundamental system of rational integral invariants of qm- If we employ not merely, as before, linear transformations with integral coefficients of determinant unity modulo y, but those of all determinants, we obtain at once the classes Cr,^x, Co (r = 1, •••, m), i-m 'm 12 THE MADISOX COLLOQUItHM. and see that these are characterized by Ai, •••, Am, h- The latter therefore form a fundamental sj^stem of rational integral absolute invariants. * But Z> is a relative invariant. General Theory of Modular Invarl\nts, §§ 8-14 8. Definitions. — Let S be any system of forms in a:i, • • • , x„ wdth undetermined integral coefficients taken modulo p, a prime. Let G be any group of linear transformations on xi, • ■ ■ , x, with integral coefficients taken modulo p. The particular systems S', S", • • • , obtained from S hy assigning to the coefficients particular sets of integral values modulo y, may be separated into classes Co, Ci, ■ • • , C„_i such that two systems belong to the same class if and only if they are transformable into each other by transformations of G. A single-valued function of the coefficients of the forms in the system S is called an invariant of S under G if, for z = 0, 1, • • • , n — 1, the function has the same value Vi for all systems of forms in the class Ci. ^ In case the values taken by are integers which may be N reduced at will modulo j) and cpngruent values be identified, the invariant is called modular. Since this reduction can be effected on each coefficient of the modular forms comprising our system S, any rational integral invariant of S is a modular invariant. An example of a non-modular invariant is the transcendental function r defining the rank of the determinant of the modular quadratic form g^. The values of r are evidently not to be identified when merely congruent modulo p. However, the residue of r modulo y is a modular invariant, since (17) r = A,- + 2A.} + • • • + mAJ^ (mod p). 9. Modular Invariants are Rational and Integral. — Any modular invariant of a system S of modular forms can be identified with a rational integraL function (with integral coefficients) of the coefficients Ci, • • • , Cg appearing in the forms of the system S. INVARIANTS AND NUMBER THEORY. 13 For, if = '^ei, .... e. when ci^ Ci, '", Cs = Cs (mod p), then (f> is identically congruent (as to Ci, • • • , Cs) to p — J s (18) Z re,....,e^Il{l-(C;-ei)^'], as shown by Fermat's theorem. 10. Characteristic Modular Invariants. — The characteristic in- variant 7^- of the class Ck is defined to be that modular inva- riant which has the value unity for systems of forms of the class Ck and the value zero for any of the remaining classes. For example, for a single quadratic form qm, Iq is given by (15),' while the characteristic invariants for the classes Cr, i and Cr, -1 are (19) Ir,l-hUr--\- Ar), I r, -1 = i{A/ - Ar) . ^ For any system of forms with the coefficients ci, • • • , (?«, we have (20) h-Zllil- (c.-c,^^>)^M, i=l where the sum extends over all sets of coefficients Ci^''\ - • ■ , c^^^^ of the various systems of forms of class Ck- In particular, in accord with (15), (21) /o = n (1 - cr')- 1=1 11. Number of Linearly Independent Modular Invariants. — Since aiiy modular invariant / takes certain values Vq, • • • , i'„-i for the respective classes Co, • • • , Cn~i, we have (22) I = Voh + l\Il + h Vn-lln-l. Hence any modular invariant can be expressed in one and but one way as a linear homogeneous function of the characteristic invariants. Moreover, the number of linearly independent modular invariants equals the number of classes. 14 THE MADISON COLLOQUIUM. For example, using (19), we see that a complete set of linearly independent modular invariants of the quadratic form q^ modulo p {p > 2) is given by (23) 7o, Ar, Ar' ir=l,'--,m- 1), D^ (k=l, ..., p-l), 12. Fundamental Systems of Modular Invariants. — ^^^lile, by (22), the characteristic invariants Iq, • • - , In-i form a fundamental svstem of modular invariants of a svstem S of modular forms, it is usuallv much easier to find another fundamental svstem. In fact, certain invariants are usually known in advance, e. g., the invariants of the corresponding system of algebraic forms. We shall prove the following fundamental theorem: 7/ the modular invariants A, B, • • ■ , L completely characterize the classes, they form a fundamental system of modular invariants. For example, 7o, • • • , In~i evidently completely characterize the classes and were seen to form a fundamental system. Let Ci, • • •, Cs be the coefficients of the forms in the system (ci, • •', Cs) ^ P(A, B, •■•, L) (mod p) for all sets of integral values of Ci, • • •, Cs. In view of Fermat's theorem, we may assume that each exponent in <^(ci, •••, Cj) is less than jj- If we replace A, • • ■ , L hy their expressions in INVAEIANTS AND NUMBER THEORY. 15 terms of the c's, P (A, • • • , L) becomes a polynomial, which, after exponents are reduced below p, will be designated by \p (ci, • • • , Cg) . Then and \J/ are identically congruent in Ci, • • • , Cs, that is, corresponding coefficients are congruent modulo i?. In fact, a polynomial of type is uniquely determined by its values for the p* sets of values of Ci, • • •, c^, each chosen from 0, 1, • • •, p — 1 (§9). Hence can be expressed as a polynomial in .1, • • • , L with integral coefficients.* 13. Minor Role of Modular Covariants. — In contrast with the case of algebraic forms, the classes of modular forms are com- pletely characterized b}' rational integral invariants. Such invariants therefore suffice to express all invariant! ve properties of a system of modular forms. In this respect, modular co- variants play a superfluous role. For instance, a projective property of a system of algebraic forms is often expressed by the identical vanishing of a covariant. But if C is a modular covariant with the coefficients ci, • • •, Cs, then 7o given by (21) is a modular invariant of C and hence of the initial system of forms. We have C = or C ^ (mod p) identically, according as 7o = 1 or 7o = 0. 14. References to Further Developments. — This general theory of modular invariants has been applied by me to determine a com- plete set of linearly independent modular invariants of q linear forms on m variables,! and a fundamental system of modular invariants of a pair of binary quadratic forms and of a pair of binary forms, one quadratic and the other linear.| The theory has been extended to combinants and applied to a pair of binary quadratic forms. § * This correct theorem for any finite field cannot be extended at once to any field as stated by me in Avierican Journal of Mathematics, vol. 31 (1909), top of p. 338. t Proceedings of the London Mathematical Society, ser. 2, vol. 7 (1909), pp. 430-444. t American Journal of Mathematics, vol. 31 (1909), pp. 343-354; cf. pp. 103-146, where a less effective method is used. § Dickson, Quarterly Journal of Mathematics, vol. 40 (1909), pp. 349-366. LECTURE II SEMIXVARIANTS OF ALGEBRAIC AND MODULAR BINARY FORMS Introductory Exl\mple of the Binary Quartic Form, §§ 1-6 1. Comparative View. — Let the forms / = ao.T^ + 4ai.T^?/ + Qaix-y- + ■^azxy^ + a^y^, with real or complex coefficients, be separated into classes such that two forms / are transformable into one another by a trans- formation of type (1) X = x' + ty', y = y', if and only if they belong to the same class. Then a single- valued function »S(ao, '•, 04) is called a semin variant of/ if it has the same value for all of the forms in any class. By the repeated application of this definition and without the aid of new principles, we shall obtain a fundamental system of rational integral seminvariants of /, then on the one hand the additional single-valued seminvariant needed to form with these a fundamental system of single-valued seminvarints, and on the other hand the additional rational integral modular seminvariants needed to form with them a fundamental system of modular seminvariants of /. It is such a comparative view that we desire to emphasize here. In later sections, we shall show that it is usually much simpler to treat the modular case independently and in particular without introducing all of the algebraic semin- variants, which become very numerous and most unwieldy for forms of high degree. The rational integral seminvariants S of an algebraic form are of special importance since each is the leading coefficient of one and but one covariant, which can be found from S by a process of differentiation. For example, the seminvariant ao is the leading coefficient of the covariant /. 16 INVAEIANTS AND NUMBER THEORY. 17 2. The Classes of Algebraic Quartic Forms. — Consider a quartic form / in which ak is the jBrst non-vanishing coefficient. Apply transformation (1) with We obtain a form having zero in place of the former ak+i. Drop- ping the accents on x', y', we obtain, for ^' = 0, 1, 2, 3, the re- spective forms (3) ao + 0: a^x^ + Qa^-^S^xY + "^a^-^SzX^ + ar^S,y\ (4) ao = 0, ai 4= 0: Aaia?y + aC^Suxf + ar^Sui/, (5) ao = Oi = 0, 02 =t= : • Qa^x^y"^ + ^ao-^S^iy^, (6) ao = ai = 02 = 0, as =1= 0: 4a3a:?/^, (7) ao = ai = a2 = as = 0: aiy^, no transformation having been made in the last case. Here (8) ^2 = aoao — a-c, S3 = ao^as — 3aoaia2 + 2ai^l (9) Si = ao^tti — 4ao-aia3 + Qaoai^a2 — 3ai^ Si3 = 4aia3 — 3a2^ Su = ai^ai — 2aia2az + a2^ 024 = 00204 — 203^. If we apply to one of the forms (3)-(6) a transformation (1) with i 4= 0, we obtain a form having an additional (second) term. Hence no two of the forms (3)-(7) can be transformed into each other by a transformation (1), so that each represents a class of forms. For example, there is a class (5) for each set of values of the parameters 02 and *S24 («2 + 0). 3. Rational Integral Seminvariants of an Algebraic Quartic. — First, oo is a seminvariant since it has a definite value 4= for any form in any class (3) and the value zero for any form in any class (4)-(7). Next, S^, S3, S^ are seminvariants, since they have constant values (11) ^2 = - Oi2, S3 = 2ai\ Si= - 3oi^ (if Oo = 0) 3 y 18 THE MADISON COLLOQUIUM. for any form in any class (4)-(7), and constant values for any form of a definite class (3), for which therefore cto has a definite value 4= and aQ~^S2, • • • , and hence each Si, has a definite value. Moreover, these seminvariants ao, S2, Ss, Si completely characterize the classes (3). Consider a quartic form / in which ao, ai, ao, as, ai are arbi- trary, except that Oo =1= 0. Any rational integral seminvariant S(ao, • • • ,ai) has the same value for / as for the particular form (3) in the same class as /. Hence S = S [ ao, 0, -^ , ~2 ' 'A I = \ ao ao ao / (f)(ao, S2, Sz, S4) ao' where ^ is a rational integral function of its arguments. We therefore seek such functions 0, we may evidently assume that not every term of the polynomial ^V in the arguments (18) has the factor gq. Let P{S2, Sz, I, J) denote the aggregate of the terms of N not involving ao explicitly. We shall prove that, if k > 0, N/ao'' is then not a rational integral function of ao, • • • , tti. For, if it be, P vanishes when ao = 0. By (11) and (16), the terms independent of ao in J involve a^, while those in I, (S2, Sz do not. Hence J does not occur in P. Then, by (11) and the term 3a2^ in I, we conclude that I does not occur in P. Thus P is a polynomial in S2 and Sz of degree or 1 in Sz and is not identically zero. By (11), it cannot vanish for ao = 0. Under the initial assumption that ao + 0, we have now proved that any rational integral seminvariant S equals a polynomial in the functions (18). The resulting equality is therefore an identity. The seminvariants (18) form a fundamental system of rational integral seminvariants of the algebraic quartic form* They are connected by the relation, or syzygy, (15). 4. Invariantive Characterization of the Classes. — By § 3, the classes (3) are completely characterized by the seminvariants ao, S2, Sz, L These with J characterize the classes (4) having ao = 0, oi 4= 0. For, by (11), *S2 and ^3 determine ai; while, by (16), / and J determine the remaining parameters in (4). * The above proof differs from that by Cayley in minor details and in the method of obtaining the functions (18) and the verification that they are seminvariants (the present method being based upon the classes). 20 THE MADISON COLLOQUIUM. The parameter aa (02 4= 0) in (5) is determined by I and J, in xdew of (17). We have now gone as far as is possible in the characterization of the classes by means of rational integral seminvariants S, since the parameters 0), of degree I + 2m and weight 21 + Qm. INVARIANTS AND NUMBER THEORY. 21 We shall make frequent use of the abbreviation (19) P. = (1 - ao^^)(l - ai^i) • . . (1 - ar'). Then PiS2i, Pidz and Pza^ are seminvariants* since each takes the same value for all forms in any class. For the classes (5), (6)> (7), their values are 3. Seminvariants' of a Modular Binary [Form of Order n, §§ 7-13 7. Fundamental System of Modular Seminvariants Derived by Induction from n — 1 to n. — It is necessary to distinguish the case in which the modulus p is prime to n from the case in which p divides n. Binomial coefficients for the form are not per- missible in the second case and often not in the first case (for example, if n = 4, p = 3, since (2) is then divisible by p). Denote the form by (20) Fn = A,x- + Aix^-'y + • • • + Any\ First, let p be prime to n. For Aq 4= 0, we can transform Fn into a form lacking the second term and having as coefficients the quotients of 0-2 = nAQA2 — 1(71 — \)A-c, ^^^^ 2, (24) Ao, 20-2 = 4:AoA2 - Ai\ PoAi, P^A^ form a fundamental system of modular seminvariants of F2. For /2, these are (24') 2ao, S2 = 00*2 — ai^, Pqcii, Pia2. 8. Order a Multiple of the Modulus. — Next, let n = pq. By Fermat's theorem, x^ — xy^~^ and hence (25) (j) = Ao(xP - xyp-^)^ is unaltered modulo p by any transformation (1). Hence if, for each value of the semin variant Aq, we separate the forms (26) Fn-i ^l(Fn- 4>) INVARIANTS AND NUMBER THEORY. 23 into classes under (1), multiply each form by y and add 0, we obtain the classes of forms F„ for this value of Aq. Hence, if n is divisible by p a fundamental system of modular seminvariants of Fn is given by Ao and a fundamental system for Fn~i. For example, if n = y = 2, Fi = {A, + A,)x + Aoy ^' can be transformed into x or Aiy by (1), according as ^o + ^i — 1 or (mod 2). Adding = A^ix^ — xy) to xy and A^y"^, we obtain representatives of the classes of forms F2. Hence the 6 classes are completely characterized by the seminvariants Ao and those (§ 7) of Fi, and hence by (27) ^0, ^1, J = (1 + ^0 + Ai)Ai. 9. Seminvariants of the Binary Cubic Form. — The classes of algebraic forms fz are (28) a,x^ + ^a^-'Sixy'' + a^-^Szf, (29) Zaix^y + \ar^Sizf, Za^xy"", a^f, where the *S's are given by (8) and (lOi). The discriminant D of/3 is given by (13). As in § 3, ao, 'S2, Sz, D form a fundamental system of seminvariants of /a; they are connected by the syzygy (13). Henceforth, let the coefficients of /a be integers taken modulo p, the excluded case p = 3 being treated in § 15. If p > 3, the classes are again (28) and (29), and a fundamental system of seminvariants is given by (30) ao, S2, S3, D, Piao, P^az. It is instructive to compare this result with that obtained by the method of § 7. Forming the functions (24) for /2' = Po/3/2/ = 3Poaia:2 + ?>Poa2xy + Poas/ (mod y), and deleting the factor 3 from the first and second, we get* Poai, 5 = Po(4aia3 — ^at^) = P^Su, Pia%, P^az. * They characterize the classes (29) of /s with Co = and may be so derived. 24 THE MADISON COLLOQUIUM. Hence, if p > 3, these four functions and uq, S2, S3 form a funda- mental system of modular seminvariants of /s. We may drop Potti since (31) PqS2 ' Ss=± 2Poai^ = ± 2Poai (mod p). Hence a fundamental system of seminvariants of /s for 2? > 3 is (32) ao, S2, Ss, d = Po5i3, Pia2, P2az. It is easy to deduce 5 from the old set (30), and D from this new set.* Finally, let p = 2. By § 7, a fundamental system of sem- invariants for fz is given by a^, S2, S3 and a fundamental system for /2'. The latter system is derived from (27) by replacing Ao, Ai, A2 by Potti, Poa2, P0O3, and hence is (1 + ao)ai, (1 + ao)a2, (1 + flo)(l + ai + a2)a3. We may drop (1 + 00)^1 — (1 + ao)S2. 10. The Binary Quartic Form. For p = 2, we have F3 = Aio^ + (^0 + A2)x'y + A3X2/ + A,f, whose seminvariants are obtained from those of /a at the end of § 9. They with Ao give a fundamental system of seminvariants of P4: Ao, Ai, A1A3 + Ao + A2, (1 + Ai)A3, AiA, + AiAziAo + A2), K = (1 + ^0(1 + ^0 + ^2 + A3)A,, An equivalent fundamental system isf Ao, Ai, A2 + A3, (l + Ai)A2, (33) ^1^44 + A0A2 + A2A3, K. * D = aoP-'iSi" + 45/) - dS2 (mod p). For, if Oo 4= 0, then 5^0 and this relation follows from (13); while, if ao = 0, D = Ui^Siz = ai^S = — Sib. Conversely, 5 can be expressed in terms of the functions (30). The above relation gives 5^2. The product of this by S^p'^ is congruent to 5 if (S2 + 0. Also 5 = if co + 0. There remains the case in which Si = 0, Oo = 0, whence ai = 0, 5 = — Saa^ = — 3(Fia2)^. t Annals of Malhemalics, ser. 2, vol. 15, March, 1914. I there give also a complete set of linearly independent invariants and of linear covariants INVARIANTS AND NUMBER THEORY. 25 For p > 3, fz is obtained from fz by replacing Qq, ai, a^y az by 4aiPo, 2a2Po, 3«3-Po, ^4^0, respectively. Making this replacement in the second set of sem- invariants of fz in § 9, we obtain P^ai, which may be dropped in view of (31), and the last five functions (34). Hence, /or p > 3, a fundamental system of modular seminvariants of f^ is given by (34) floj S2, Sz, Si, PqSiz, PoSii, P1024, P2O3, Pzdi' Here the three Sij are given by (10). Since the functions (34) completely characterize the classes (3)-(7), we have a new proof that they form a fundamental system. 11. Explicit Fundamental System ivhen p > n. — Instead of employing the above step by step process, we can obtain directly a fundamental system of modular seminvariants of fn when the modulus p exceeds the order n of the binary form (22). Consider a particular /n in which ak is the first non-vanishing coefficient: Z ( ^ ) ttiX'^-y (flk + 0). i=k \ * / To this we apply transformation (1) and obtain i=k j=0 \ ^ / \ 3 J l=k where we have replaced j by I — i and set -"=s(:)C;-";)-''--(:)s(9-'-- Take k < n and give to t the value (2). Thus (n\ (35) "^'^ " {{k+l)auV-' ' 26 THE MADISON COLLOQUIUM. In particular, (^kk — Ij <^kk+\ = 0, k). It follows also that the parameter ak+\ in a class Ck+i is de- termined by the semin variants Pk-iO'ki (I = k -{- 2, k -{- 3), provided k -{- 3 ^ n. But a„_i and a„, not so determined, are found from Pk-idk (k = n — 1, n). Hence a fundamental system of modular seminvariants of fn, for p > n, is given by ao, (Xoi (/ = 2, • • •, n), (36) Pk-KTki {k=l,---,n-2',l = k-\-2, ■'•,n), Pn—20'n—l, Pn—lttri' For 71 = 2, 3, 4, these are (24'), (32), (34), respectively, except for the difference of notation indicated above. For n = 5, we see that a fundamental system of modular seminvariants of /s, for p > 5, is ■\ ao, S-z, S3, 04, 05, PqSis, PoSu, P0S15, (37) PlS2i, PlS2i, P2S35, Psdi, P^db, INVARIANTS AND NUMBER THEORY. 27 in ivhich the symbols are defined by (8)-(10), (19) and /So [= ao%5 — 5ao^aiai + 10ao^«i^«3 — 10aoai^a2 + 4a i^ *Si5 = 16a 1^05 — 40ai%2a4 + 40aia2^a3 — 15a2^ (38) Sih = 27a2-a5 — 45a2a3a4 + 20a3^ Sz5 = SasOs — 5a4^ 12. Another Method for the Case p > ??. — We may formulate the method of § 7 so that it shall be free from the induction process. The classes of forms (23) with PoAi 4= 0, and hence the classes of forms Fn with .4o = 0, Ai ^ 0, are characterized by the seminvariants given by the products of Po by the functions az', • • ' obtained from 0-2, 0-3, • • • , (Xn-i by increasing the subscript of each Ai by unity and replacing nhy n — 1; indeed, Pi^ = Pi (mod p). When the process of deriving (23) from (20) is applied to (23), we get F'U = [1 - (PoA,)^']F:_Jy = (1 - A,^-')PoFjf (39) = PiFnhf ^ PiA^x^-' + PiAzx--'y H hPi^n?/"-' (mod 2?). The class of forms (39) with P1A2 + 0, and hence the classes of forms Fn with Ao = Ai = 0, A2 4= 0, are characterized by the seminvariants given by the products of Pi by the functions 0-2", • • • obtained from a^, • • • , (rn-2' by increasing the subscript of each Ai by unity and replacing nhy n— 1. Finally, we obtain P„_2.4„_i.r + Pn-2Any, characterized by the seminvariants Pn-2^n-i and Pn~iAn. The earlier Pk-iAk may be dropped (§ 11). For example, if n = 3, p > 3, the fundamental system of F3 is Ao, a2, as, Po o, we take ao, S2, S3, Si and the products of Po by the functions f n and employ the notations of § 1 1 . In the classes Ck (k n. Deeivatiox* of ^Modular Invariants from Seminvariants, §§ 14-15 14. Invariants of the Binary Quadratic Form. — First, let p=2. Any pol\'nomial in the seminvariants (27) is a linear function of 1, Aq, Ai, AqAi, J, AqJ = A0A1A2, since (^0 + Ai)J = 0. Since there were six classes, these six seminvariants form a complete set of linearly independent sem- invariants. Now a seminvariant is an invariant if and only if it is symmetrical in ylo and .42. But I={l-Ao)a-Ai)(l-A2)^(l-Ao)(J+l+A,) (mod 2). Thus 1, Ai, AqJ and / are invariants. By subtracting constant * WMle this method is usually longer than the ntiethod of Lecture I, it requires no artifices and makes no use of the technical theory of numbers. Moreover, it leads to the actual expressions of the invariants in terms of the seminvariants of a fundamental system, thus j-ielding material of value in the construction of covariants. INVARIANTS AND NUMBER THEORY. 29 multiples of these four, any seminvariant can be reduced to cAo + dAoAi, which is an invariant only when identically zero. Hence 1, Ai, A0A1A2 and I form a complete set of linearly inde- pendent invariants of F2 modulo 2. Next, let p > 2. The discriminant of f2 is D = 82- Any polynomial in the four fundamental seminvariants (24') is a linear function of ao'D^', Poai\ Pia2' {i, i = 0, 1, • • •, p - 1), since the product of PoQi or PiOa by ao is zero, that of Pia2 by PoQi or D is zero, while DP oai = — Pai^. Further, Po = 1 - ao^\ PoW^' - (- am = 0, Pi = Po - Poai^\ ao^'D^ ^ D^' - (- l)^Poai'', modulo p. Hence any seminvariant is a linear function of ao^\ aoW^ (i=0,l, •■-,p-2;j = 0,l, '■',p-l), ^^^^ Poa,\ Pia2^ {k=l, '",p-l). The number of these is p^ + p — 1. Hence (§ 13) they form a complete set of linearly independent modidar seminvariants of fi for p> 2. The invariant ^ = yli in § 6 of Lecture I becomes for two variables (41) ^={ao" + a2"(l-ao^^)}(l-Z>^^) = ao"(l-^^')+^i«2% where /z = (p — l)/2. By the expansion of Dp~^, we get* (42) A = (flo'^ + 02'') ( 1 - I^ a^Wai^'^-'^'J . * Transactions of the American Malhemalical Society, vol. 10 (1909), p. 132. To give a direct proof of the identity of the final expression (41) and (42), note that the product of the final factor in (42) by D equals OoOz — {aoa2)i^'^^ algebraically, so that the product AD is divisible by p. But the product of (41) by Z) is evidently divisible by p. It therefore remains only to treat the case D =0. Replacmg ai^ by aoa2, we see that the final factor in (42) becomes I — (jj. + l)ao'^C2*'. Hence (41) and (42) are now identical if ao'^aa" (00** — 02*^) = (mod p). But, if aoUi =t= 0, Oo'^Cz'^ = Ci^*^ = 1, ao*^ = azi^ = ±1. 30 THE MADISON COLLOQUIUM. Since (42) is therefore a seminvariant and is symmetrical in ffo and a2 and since the weight of every term is divisible by p — 1, ^ is an absolute invariant. By (41), A" = flo"^ (1 - D^^) + Pla22^ (1 - flo^')Z)^i = Poai^\ (43) A^ + D^'-l^-Io, /o=(l-«o^')(l-ai^')(l-a2^'). Hence also Iq is an absolute invariant. Subtracting multiples of I,= l-ao^'-Poai^'-Piao^\ A, D^ {j=0, 1, • • •, p-1), we may reduce any seminvariant to a linear function of the ex- pressions (40) other than Pirt2^', Pia2'', D^' (j = 0, - • -, p — 1). The resulting linear function L is not an invariant. For example, if 2? = 3, it is L=aao'^-]-bao-{-caoD-\-daoD~-{-ePoCii-{-fPo(ti' {a, ' • •,/ constants). Interchange ao and 02, and change the sign of Ui. We get aa2^ + ba2 + ca2D + c?a2^^ + (1 — 02") (/a 1^ — eai). This is to be identically congruent to the invariant L. Taking 02 = 0, we see that e=f=a=b = 0, c = d. Then L = caQa2(ao + a2) + ca^a-^a2 is not symmetric in Qq and 02. Hence i = 0. For any p, a like result may be proved by con- sidering separately the terms of L of constant weights modulo p — \. Hence in accord with § 11 of Lecture I, a complete set of linearly independent invariants of /2, for p > 2, is given by Iq, A and the powers of D. In place oi D^ = 1 , we may use A^, in view of (43). 15. Invariants of the Binary Cubic Modulo 3. — A fundamental system of seminvariants of Fz modulo 3 is given by Aq and a fundamental system of F2 = Aix' + (Ao + A2)xy + Azif. Hence, by (24), a fundamental system for F3 is given by Ao, Ai, t = AyAz - (Ao + ^2)-, (1 - Ai')(Ao + A2), u= (l-^i2)[l- (^0 + .•l2)-M3. nn'AEIAXTS AND NUMBER THEORY. 31 In place of the fourth and third we may evidently use X = (1 - ^li-)^o, 0- = ^1^13 + A0A2 - AM2' = i + ^0' + v. Here a is the discriminant of F3 for p = 3. By § 13 there are 11 classes of forms F^. Hence, by §8, there are 3-11 classes of forms Fz. Thus there are exactly 33 linearly independent seminvariants of F3. Since A{\ ^ Aifx =0, (tX = ^oX', mCo- + ^0') = 0, ;,(X + ^0) = 0, (1 - Ai^)(r = .4oX, modulo 3, any polynomial in the seminvariants Aq, A\, a, X, /x of the fundamental system is congruent to a linear function of (44) Aq^Ai\ Aq'(j\ Ao'Ai<7\ /lo^V, ^oV (i, j=0, 1, 2; k= 1, 2). Hence these 33 functions form a complete set of linearly inde- pendent seminvariants of Fz. The seminvariants P = 1 - ^1^ _ x2 = (1 _ A,')(l - A2'), (45) 7o = (1 - Ao')(P - M^) = n (1 - A,'), t=0 E = AoAi( 2 (rele- gating to foot-notes the modifications to be made when p = 2). Then there are two transformations (3). Hence any non-special point is one of exactly* (4) CO = ip(p2 - 1) conjugate points under the group G, while a special point is one of fewer than w conjugates. Let {x, y) be a special point and let (1) be a transformation, not of the form (3), which leaves it invariant. Thus the con- gruences (2) are not both identities. The determinant of their coefficients must therefore be divisible by p. Hence p is a root of the characteristic congruence (in which a = 6 + e) (5) p^ — ap + 1 = (mod p). First, suppose that (5) has an integral root p. For this value of p, one of the congruences (2) is a consequence of the other, and the ratio a: : ?/ is uniquely determined as an integer modulo p. * For p = 2, w is to be replaced by 2(2^ — 1) =6. INVARIANTS AND NmiBER THEORY. 35 Hence only real special points are invariant under a transforma- tion [other than (3)] whose characteristic congruence has an integral root. Moreover, all real points are conjugate under the group G. Indeed, x' = bx, y' ~ x-i- b~^y, and x' = — y, y' ^ x replace (1, 0) by {h, 1) and (0, 1) respectively. Hence if an invariant of G vanishes for one of the real points, it vanishes for all and has the factor p-i (6) X = 2/ n (--c - ay) = x^y - xy^ (mod p), a=0 the congruence following from Fermat's theorem. Obviously, any transformation of G replaces a real point by a real point, and therefore L by kL. The constant k is in fact unity and L is an invariant of G. Indeed, for (7) x^aX+hY, y^cX+dY (mod p), where a, ''•, modulo p- Making use of the theorem in § 2, 1 have proved by inductionf that the m invariants Lm, Qmi, • - ', Qmw-i are independent and form a fundamental system of rational integral invariants of r^. A fundamental system of invariants of the group of all modular linear transformations on two sets of two cogredient variables has been obtained very recently by Dr. W. C. Krathwohl in his Chicago dissertation. t Formal Invariants and Seminvariants of Modular Forms, §§ 5-13 5. Formal Modular Invariants. — Consider a binary form f(x, y) = aQX" + aix'-hj + • • • + ary\ in which x, y,ao, • • • , ar are arbitrary variables. The transforma- tion (7) with integral coefficients, whose determinant A is not divisible by the prime p, replaces / by a form 0(Z, Y) = AoX^ + AiX'-'Y + • • . H- ArY\ in which (18) Aq = /(a, c), Ai = ra'-^bao + • • • , • • • , Ar = fib, d). A polynomial P(ao, • • • , a^) with integral coefficients is called a formal invariant modulo p of index X of / under the transforma- * E. H. Moore, Bulletin of the American Mathematical Society, vol. 2 (1896), p. 189. His proofs do not use the invariantive property. A lilce remark is true of the proof that the product (17), in the case a;m = 1, is congruent to a determinant of order w — 1, then obviously equal to Lm, by R. Levavasseur, Memoires de I'AcadSmie des Sciences de Toulouse, ser. 10, vol. 3 (1903), pp. 39-48; Comptes Rendus, 135 (1902), p. 949. t Transactions of the American Mathematical Society, vol. 12 (1911), p. 75. J American Journal of Mathematics, October, 1914. INVARIANTS AND NUMBER THEORY. 41 tion (7) if (19) P(Ao, Ai, ■•-, Ar) = A^P(ao, a,, - - ■ , Or) (mod v), identically as to ao, • • •, Or, after the A's have been replaced by their values (18) in terms of the a,. If P is invariant modulo p under all transformations (7), it is called a formal invariant modulo p of /. The term formal is here used in connection with a form / whose coefficients are arbitrary variables in contrast to the case, treated in the earlier Lectures, in which the coefficients are undeter- mined integers taken modulo j). In the latter case, (19) neces- sarily becomes an identical congruence in the a's only after the exponent of each a is reduced to a value less than p by means of Fermat's theorem a''^ ^ a (mod p). The functions (18) are linear in oo, • • • , Gt. It is customary to say that relations (18) define a linear transformation on ao, • • • , Or which is induced by the binary transformation (7). Let V be the group of all of the transformations (18) induced by the group of all of the binary transformations (7). Making no further use of the form /, we may state the above problem of the de- termination of the formal invariants of / in the following terms. We desire a fundamental system of invariants of group V. This problem is of the type proposed in § 1; the group F is a special group of order a multiple of p. Here and below the term in- variant is restricted to rational integral functions of ooj • • • j cfr. A theory of formal invariants has not been found. For no form / has a fundamental system of formal invariants been published. Some light is thrown upon this interesting but difficult problem by the following complete treatment of a binary quadratic form, first for the exceptional case p = 2 and next for the case p > 2, and preliminary treatment of a binary cubic form. 6. Formal Invariants Modulo 2 of a Binary Quadratic Form. — Write (20) S=ax'+ hxy + cy\ 42 THE MADISON COLLOQUIUM. where a, h, c are arbitrary variables. Under the transformation (21) x = x' + y', y = y', f becomes /', in which the coeflBcients are (22) a'=a, h' ^ h, c' = a + h ^- c (mod 2). By § 3, the only invariants under d' = d, c' ^ c -\- d, modulo 2, are the polynomials in d and c(c + d). Take d = a-\- h. Hence the only seminvariants of f are the polynomials in a, b and (23) s = c(c + a + b). Such a polynomial is an invariant of / if and only if it is unaltered by the substitution (ac) induced by (xy). Thus (24) b, k = as, q = b(a ■}- c) + a^ + ac -{• c^ = s -\- ab -{- a^ are invariants of /. Introducing q in place of s, we see that any seminvariant is a polynomial in a, b, q. Consider an invariant of this type. Since its terms free of a are invariants, the sum of its terms involving a is an invariant with the factor a and hence also the factors c and a -}- b -{- c, the last by (22) . Hence this sum has the factor k, and its quotient by k is an invariant. By induction we have the theorem: Any rational integral formal invariant of f equals a rational integral function* of b, q, k. 7. Formal Seminvariants of a Binary Quadratic Form for p> 2. Write (25) / = ax' + 2bxy + cy\ where a, b, c are arbitrary variables. Under the transformation (21),/ becomes/', whose coefficients are (26) a' = a, b' = a+b, c' = a + 2b + c. * Replace Xi, Xz, Xz, of § 4 by a, b, c ; then L, = bk{k + bq), Qzi = h* + bk -\- q\ Qzi = 6Y + ^'2^ + h^k + k\ IXVARIANTS AND NUMBER THEORY. 43 Evident formal seminvariants are a, A = 6^ — ac, and p-i (27) ^ =^ n (ta + b) ^ bP - ba^^ (mod p), (28) T. = n {(f - k)a + 2tb + c] {k = 0, 1, • . ., p - 1). Indeed, the linear function under the product sign in (28) is transformed by (26) into the function derived from it by re- placing ^ by / + 1. As in (27), (29) [Tt]a=o = cP - c6^i (mod p). Let S{a, b, c) be a homogeneous rational integral seminvariant with integral coefficients. Then, by (26), S{0, b, c) = S{0, b, 26 + c) (mod p). Thus, by § 3, 5(0, b, c) equals a polynomial in b, c^ — cb'^^. Hence, by (29), S{a, b, c) = a(T{a, b, c) + 4>{b, jk) (mod p), where a and 2, any formal seminvariant of a binary quadratic form is a polynomial in a, A, 70 or the product of such a polynomial by jS. 8. Formal Invariants of a Binary Quadratic Form for p > 2. The product (32) r = JJ^ 7/b {k ranging over the quadratic non-residues of p) is an absolute invariant of / under the group G of all binary transformations with integral coefficients taken modulo p of INVARIANTS ANT3 NUAIBER THEORY. 45 determinant unity. It suffices to prove that this seminvariant is unaltered bv the substitution (33) a' = c, c' = a, h' = - 6,1 induced by the transformation x = y', y = — x'. ^ Under (33), the general factor in (28) is replaced by {t^-k){{T--K)a + 2Th + c}, where 7' = 7o— ^, K= ^ f-k' {f-kf Hence K is quadratic non-residue of 'p when k is. Also, 11(^2-^0= -^- In {k-e)\ ^-kilH-ir-^-^k (modp) t=Q \ t=l J if A; is a non-residue. To show^ that the product of the resulting numbers — 4^• is congruent to unity, we set a: = in p-i (34) n {x -k) = x' +1 (mod p), k and note that 2^-1 ^ 1. Hence (32) is unaltered by (33) and is an absolute invariant of / under G. It is very easy to verify that (35) J = 070 is unaltered by (33), so that J is an invariant of / under G. If an invariant has the factor /3, it has the factor (36) B = jSn7r (/• ranging over the quadratic residues of p). For, under the substitution (33), 64-ra (r=#0) becomes T{c—h!r). By choice of r, we reach c + 2ib, where t is any assigned integer not divisible by p. This is a factor of 7^ where k = f. The fact that B is an invariant may be verified as in the case of (32) or deduced from the fact that a^ n 7/t = ayo ' BT 46 THE MADISON COLLOQUIUM. is an invariant, being the product of all non-proportional linear functions of a, b, c with integral coefficients modulo p. Hence any invariant is the product of a power of B by an invariant which is a polynomial P in a, A, 70. Since jk is a seminvariant not divisible by |S, it equals a polynomial in a, A, 70 (§ 7). But if a = 0, 7^ = 70 (mod p), by (29), and A = 6^ is free of c, so that 7^ is not a polynomial in a and A only. Hence (37) 7;t = To + 9k{a, A) (mod p). For 2^ = 3, the polynomial P therefore equals a polynomial in a, A, 72 = r. Now an invariant 4>{a, A, F) differs from the invariant 0(0, A, F) by an invariant with the factor a and hence the factor (35). Treating the quotient similarly, we ultimately obtain the following theorem for the case p = 3 : A fundamental system of formal invariants of the binary quad- ratic form f modulo p, p > 2, is given by the discriminant A and F, J, B, defined by (32), (35), (36). The product of the last three is congruent modulo p to the product of all the non-proportional linear functions of the coefficients of f. To prove the theorem for 2? > 3, note first, by (37), that F, given by (32), differs from 70"^ by a polynomial in 70, a, A of degree n — 1 in 70, where n = {p — I) 12. Hence a polynomial in a, A, 7o equals a polynomial in a, A, 70, F of degree at most n— 1 in 7o. Subtract from each the terms of the latter involving only the invariants A, F. We have therefore to investigate invariants of the type TO— 1 n—1 (38) Z c^yo'P^(A, F) + E yo'(f>i{a, A, F), i=l 1=0 in which the Ci are integers, while Pi and 0i are polynomials in their arguments, and <^t has the factor a. If every Cf = 0, the invariant has the factor a and hence the factor ayo = J, and the quotient by J is an invariant which may be treated similarly. The theorem will therefore follow if we show that a contradiction INVAEIANTS AND NUMBEH THEORY. 47 is involved in the assumption that a certain Cj is not divisible by p. First, the remaining d are divisible by p. For if also c,- ^ 0, let Z-,A'"»T*»' be the term of P, of highest degree in A. Since 7o and T are of degrees p and np, and of weights = 2 and (mod p — I), yo^Pi is of degree pi + 2ri + Sinp and of weight = 2i + 2r,- (mod p — 1). But p = 1 (mod n). Hence i+2ri = i+2ry, 22 + 2ri = 2i + 2ry (mod n), so that i = j (mod n). But i and j are positive integers < n. Hence i = j. Multiplying our invariant by a suitably chosen integer, we have the invariant «-i (39) 7o^Py(A, r) + Eyo'Ua, A, F), Pj = AT« + • • -. t=0 Now — (c — ka)h^^ is the term of highest degree in h in yk- Hence (40) 70= _c6p-i+ ..., F = (r6"(P-i^+ •••, (41) a = ni- (c - ^-a)) = (- oY + (- a)" (mod p), k where k ranges over the non-residues of p, the last following from (34) for x = c/a. Since 70 and F are of even weights, only even powers of b enter (39). Hence an invariant (39) is symmetrical in a and c. We shall prove that this is not the case for the terms of highest degree in 6. For yo^Pj this term is (42) (-c)V6^ I3 = j(p-l) + 2r+sn(p-l), Let da^W^T^* be one of the terms of 0i in which the exponent of 6 is a maximum. Then in yo%i the highest power of b occurs in the terms (43) da^^i- cYaH'<, /?.■ = 2/,- + goi{p - 1) + Hv - D- Since the weight and degree is the same as for (42), 22 + /3i^2i+/3 (modp-1), (44) ei-\- 1+ QiU + jSi = ; + sn + /3. 48 THE MADISON COLLOQUIUM. First, let (3i = /?. Then i = j, ei=0 (mod n), whence i = j. Thus the exponent of a in any term (42) or (43) is divisible by n, while the exponent of c is not, being congruent to j modulo n. Hence the coefficient of b^ in the sum of (42) and the various terms (43), with i = j, is not symmetrical in a and c, unless identically zero. But (43) has the factor a while (42) does not. Hence the greatest (Si exceeds jS. Next, consider a set of terms (43) and a set of terms of like form with i replaced by k, all being of equal degree in b. Then Pi = ^k. By (44i), 2i + 0i^2k + Z?/,, i = L Consider finally terms (43) with jSf constant. In them the residue modulo n of ei is a constant =# i. For, if Ci = i, then 2i -{- j3i = j -\- j3 (mod n) by (442), so that j = (mod n) by (44i). Hence these terms (43) are not symmetric in a and c and yet do not cancel.* Our fundamental invariants are connected by a syzygy; for 2? = 3, (45) B^ = AT2 -I- J(J _ A2)2. 9. Formal Invariants of a Binary Cubic Form for p 4= 3, — We have seen that the theory of formal invariants of a binary quadratic form is dominated by the invariantive products of linear functions of the coefficients. While these products de- pended upon the classification of integers into the quadratic residues and the non-residues of p, we shall find that for a cubic form it is a question not merely of cubic residues and "non-residues of p, but of the larger classes of reducible and irreducible con- gruences. Write / = ax^ + dbx^y -\- Scxy^ + dy^, thus taking p =1= 3. Under transformation (21), / becomes /', whose coefficients are given by (26) and (46) ^' = a + 36 + 3c + ^. * If two are of like degree in c, their ^'s are equal and hence their /'s are equal; then, if of like degree in a, their e's are equal. But then we have the same term of 0f. INVARIANTS AND NUMBER THEORY. 49 Hence a, /3 and yic, given by (27) and (28), are again seminva- riants; also, p-i (47) 5y, = !!{(<'- 3A-i - j)a -\- W - k)b + Stc + d} (j,k = 0, ...,p- 1). Indeed, if Ft{a, b, c, d) is the function in brackets, Ftia', h', c', d') = Fi+iia, b, c, d). Any invariant with the factor a has the factor (48) adoo = aU {t\ + 2>fb + Ztc -{- d) = /(I, 0) n/(^, 1), whose vanishing is the condition that one of the points {x, y) represented by / = shall be one of the existing p + 1 real points (1, 0), {t, 1) of the modular line. To verify algebraically that the seminvariant (48) is an invariant,* note that it is unaltered modulo p by the substitution (49) a' = - d, d' ^ a, b' = c, c' = - b, which is induced on the coefficients oi f by x = y', y = — x'. The product P of the 8jk in ivhich j and k are such that \ = t'- m - j is irreducible modulo p is a formal invariant. The substitution (49) replaces the general factor of (47) by - a 4- 3^6- 3(^2 _ i;)c + \d = X{ (P - 3KT - J)a + 3(r - K)b + ZTc + d}, where f7 = F + kf + tj, h= - 2k^ + QkH^ + 3ktj + t'j + f. * For any form, see Transactions of the American Mathematical Society, vol. 8 (1907), pp. 207-208. 50 THE MADISON COLLOQUIUM. We are to show that there is no integral solution x of x^ - ZKx - J=0 (mod p). Multiply this by X^ and set X.t = y. Then y^ — 3gy — h = (mod p). But the negative of the left member is the result of substituting r -\- s = — t, rs = — y — 2k in the expansion of the product The latter is congruent to zero modulo p for no values of r and s which are integers or the roots of an irreducible quadratic congruence with the integral coejfficients t, — y — 2k. For p = 2, P = 5ii. For p = 5, P is the product of two invariants* (50) 5ii522532541, ^uhA^SA^AS, neither of which is a product of invariants. The last property is true also of the following invariants : 7l5o3> 74^02, T25o45l253o52o542> (51) The product of these seven invariants and aSoo equals the product of all the linear functions of a, h, c, d, not proportional modulo 5. For p = 2, each of the 15 linear functions is a factor of just one of the following invariants (no one with an invariant factor) : (52) adoo, du, fiyodou K = b + c, (a + b -\- c)5io. For any p ^ 3, the cubic form has the formal invariant (53) G = 3(6cP - b^c) - (ad^ - aH), * In those linear factors of the first which lack c, the product of the coef- ficients of a and 6 is a quadratic non-residue of 5; in those of the second in- variant, a quadratic residue. INVARIANTS AND NUMBER THEORY. 51 and an absolute formal invariant* K of degree j) — 1. For p = 5, (54) X = 6* + c^ - h'-d? - a^(? - hcH - ahh + acd? + a'hd. Thus, for p = 5, K and the discriminant D are invariants of degree 4, and weights = 0, 2 (mod 4), while aSoo and G are of degree 6 and weight = 3 (mod 4). It follows from § 10 that there are no further invariants of degree less than 8. Now the first and second invariants (51) are of degree 10 and weight = 1 (mod 4). Hence if either is expressible as a polynomial in in- variants of lower degrees, it must be the product of D by a linear function of adoo and G. This is seen to be impossible either by a consideration of the terms of degree ^ 5 in d or by noting that D has no linear factor. Thus 7i5o3 or 745o2 occurs in a fundamental system of invariants. Invariantive products of linear functions of the coefficients of the cubic form therefore play an important role in the theory of its formal invariants. Whether or not they play as dominant a role as in the case of the quadratic form is not discussed here. We shall however treat more completely the seminvariants. 10. Formal Seminvariants of a Binary Cubic for p > 3. — We shall first determine the character of the function to which any seminvariant S{a, b, c, d) reduces when a = 0. Set A = 36, 2B = 3c,C = d. Then (26) and (46) give A' = A, B' = A + B, C = A-\-2B + C (when a = 0). Any function unaltered by this transformation is (§ 7) a poly- nomial in A, B^ — AC, 70', or the product of such a polynomial by /S^ where 70' and /3' are the functions 70 and jS written in capitals. But 70' = n m'b + StC +d) = [5,o]a=0, t=0 * Transactions of the American Mathematical Society, vol. 8 (1907), p. 221; vol. 10 (1909), p. 164, foot-note. Bulletin of the American Mathematical Society, vol. 14 (1908), p. 316. Cf. Hurwitz, I. c. 52 . THE MADISON COLLOQUIUM. modulo jp. Hence (55) S = aaifl, h, c, d) + yk^4>(b, q, 5yo) (e = or 1), where k, j may be given any assigned integral values and (56) ? = c2 - ibd, - Sb'q = [D]a=o, D being the discriminant of/. We use the semin variants (II, § 2) (57) S2=-b'' + ac, Ss = 2¥-{-a{ad-Sbc). First, let p = 5. Then q = c^ -\- 2bd. We have the formal semin variants* (Tz = bq — a{ab + 2cd), ai = K- S.}= q^+ a{abd - 2ac2 + b'^c + cd^), (75 = bq^ + a{- ad^ - bed" + ^cH + abc" - 2¥c + a^b), a^= q^ + a {ad'' - 2bcd^ - cW + abcH - 2¥cd + a^bd-{- 2ac* (58) - b^c^ - 2aV + a¥)\ 2, it involves only even powers of /, and by the homogeneity, only even powers of q. Hence any absolute in' variant of T is a product of powers of L^"^ and Q by a polynomial in q"*, P, where y = lifp=2,y = 2ifp>2. In particular, L'p~^ and Q form a fundamental system of absolute invariants of F. The so-called form problem for the group F requires the determination of all pairs of values of the variables X and y for which L^^^ and Q are congruent modulo p to assigned values X and fx, either integers or imaginary roots of congruences modulo p. We have therefore to solve the system of congruences (65) {L{x,y)}^'^\ Q{x,y)^fx (mod p). * After these lectures were delivered, I saw a manuscript by Professor O. E. Glenn, containing tables of formal concomitants for forms of low orders and moduli 2 and 3. He employs transvection between the form and the covariant L of § 2. INVAEIANTS AND NUMBER THEORY. 59 First, let X ^ x^' y^ = z'P y^ X y For z = X ov z^ y, we have = Lz^' - QLz^ + L^z (mod p). Hence x and y are roots of (66) F(z) = sP' - /X2P + X2 = (mod p). Having no double root, this congruence has p"^ distinct integral or imaginary roots. These roots are (67) eX-hfY (e,/=0, 1, ...,p-l), where X and Y are particular roots linearly independent modulo p. For, (68) F{eX + fY) = eF{X) + fF{Y). Hence any pair of solutions x, y of (65) is of the form (7), where a, ' • -y d are integers, whose determinant A is not divisible by p, in view of (64i) and X ^ 0. Conversely, if A' and Y are fixed linearly independent solutions of (66), any pair of linear functions of X and Y with integral coefficients, whose determinant is not divisible by p, gives a solution of (65). Indeed, by (68), x and y are solutions of (66). From the two resulting identities, we eliminate X and ju in turn and get M = Q{x, y), {L{x, y)]p = \L{x, y). Since X and Y are linearly independent modulo p, L{X, Y) is not divisible by p [cf. (6)]. Thus L{x, y) ^ by (64). Hence (65) hold. Hence, for X ^ 0, the form problem has been reduced to the solution of congruence (66). The latter will be discussed here in the simple but typical* case in which X and fx are integers. Now the problem to find the real and imaginary roots of a con- * For the general case, see Transactions of the American Mathematical Society, vol. 12 (1911), p. 87. 60 THE MADISON COLLOQI'IOI. gruence with integral coefficients is at bottom the problem to factor it into irreducible congruences with integral coefficients. When V is an integer, z^ — vz is a factor of (66) if and only if V is a root of the characteristic* congruence (69) v^-fxv + \=0 (modi^). Such a binomial is a productf of binomials z'^ — 8, irreducible modulo p, whose degree d is the exponent to which the integer v belongs modulo p. Since 2p — 1 < p^, the function (66) has an irreducible factor 0(2) of degree D > 1, not of the preceding type z"^ — d, and hence with a root r such that r^/r is not congruent to an integer. Thus every root of (66) is of the form Cir+C2r^, w^here the c's are integers. The irreducible factors of (66) are of degree D except those, occurring only ichen (69) has an integral root, of the form z'^ — 8, where dis a divisor of D. To find D, note that by raising (66) to the powers p, p^, • • • , we can express z^' as a linear function U of z^ and z. Now D is the least value of t for which k = z. But the coefficients of It are the elements of the first row of the matrix of S^~^, where S = (ri) * Note the analogj' of (66) with the Hnear differential equation ha^^ng the solution z = e" if t; is a root oi v^ — fiv + X = 0. Also, (68) holds. Make dz/dt correspond to zp and hence dh/dt^ to {zp)p. Thus the differential equation corresponds to (66), and the integral z = e"' (viz., dz/dt = vz) to gP = vz. t Let f{z) be an irreducible factor of degree d. Its roots are r, rP = vr, r^ = v^r, • • -, rP ~ = v^'h, where v^ = \, v^ ^ \, Q < I < d. Thus d is a divisor of p — 1. Hence 2P-1 — V = zP-'^ — rP-^ has the factor z'^ — r''. The latter has a root r in common with /(z). But (H)p-i = v"^ = \. Thus 5 = r'' is an integer. Hence f{z) = z'' — 8. im^ARIANTS AND NUMBER THEORY. 61 But Id = z implies that Id+i = z^. The condition for the latter is therefore S^ = 1. Hence D is the period of S. But (69) is the characteristic determinant of S. According as it has distinct roots Vi and v^ or equal roots v = ^fx = \^, a linear substitution of matrix S can be transformed linearly into one of matrix* \0 vj' \0 v)' According as the characteristic congruence (69) has distinct {real or imaginary) roots or a double root, D is the least common multiple of the exponents to ichich the distinct roots belong modulo p, or is p times the exponent to which the double root belongs. Finally, let X = 0. By (6), either y = or x — ay = (mod p), where a is an integer. In the first case, Q = xP~P, x^"- - fxx^ = 0. If /x = 0, then x = y = 0. If /i 4= 0, the roots x are equal in sets of p and hence are cxi (c = 0, 1, • • •, p — 1), where xi is a particular root not divisible by p. In the second case x — ay^O, we take a; — a?/ as a new variable X and conclude from the absolute invariance of Q that Qix, y) = Q{0, y) = 2/^'-^ We thus have the first case with y in place of x. Using similar methods, I have solved the form problem for the total group of modular linear transformations on m variables. f 15. Invariantive Classification of Forms. — ^Let (70) 0(.T, 2/) = a:«+ ••• (m > 1) be a binary form irreducible modulo p and having unity as the coefficient of the highest power of x. Let G be the group of all modular binary linear transformations (1) with integral coef- * In the second case we use the new variables x and x — vy. t Transactions of the American Mathematical Society, vol. 12 (1911), pp. 84-92. 62 THE MADISON COLLOQUIUM. ficients of determinant unity. Let (f)i = 4), 4>2, • • • , 4)k denote all the forms of type (70) which can be transformed into constant multiples of ^ by transformations of G. Evidently their product P = 0102 • • • 4>k is transformed into CtP by any transformation t of G. The constant Ct is easily seen* to be congruent to unity. Hence P is an absolute invariant of G. If m > 2, no 0; vanishes for a special point. We now apply the theorem in the first part of § 14. Hence, if m > 2, the absolute invariant P is an integral function with integral coefficients of the invariants q, I, each ex- ponent of q and I being even if p > 2. In view of the definition of the (f)i, this function of q and / is an irreducible function of those arguments modulo p. Two binary forms shall be said to be equivalent if and only if one of them can be transformed into a constant multiple of the other by a transformation of G. A set of all forms equivalent to a given one shall be called a genus. Thus 0i, - ' ■ , 4>k form a genus. All of the irreducible forms (70) separate into a finite number/ of distinct genera; let Pi, • • •, P/ denote the products of the forms in the respective genera. Thus -Km = P\ • • • Pf is the product of all of the binary forms x^ -\- • • • irreducible modulo p. Hence iTm is a polynomial in q, I with integral coef- ficients. Hence the f genera of irreducible binary forms of degree m > 2 are characterized invariantively by the f irreducible factors Pi(5'j I) of Tmiq, I) modulo p. We shall see that Xmiq, I) is easily computed. By finding its factors irreducible modulo p in the arguments q, I, we shall have invariantive criteria for the equivalence of two irreducible binary forms of degree m. For example, we shall prove that TTz = q — I ii p = 2, so that all irreducible binary cubic forms modulo 2 are equivalent. Further, tts = q"^ — F U p > 2, so that the irreducible cubic factors oi q — I are all equivalent, also those oi q -\- I, while no factor of the former is equivalent to one of the latter. * Transactions of the American Mathematical Society, vol. 12 (1911), p. 3, § 4. The present section is an account of the simpler topics there treated at length. INVARIANTS AND NUMBER THEORY. 63 In general, let m be a product of powers of the distinct prime numbers qi, • • •, q^, and set From the expression for 7r,„ due to Galois we readily obtain ^ _ Fm ■ UFmlqjgj • H/^m/g^g.gtgt ' ' ' UFmiQi ' UFmigigjg^ ' " in which the first product in the numerator extends over the 2M(m — 1) combinations of gi, • • •, q^ two at a time, and similarly for the remaining products. By the first theorem of this section, and (11), TTm is a polynomial in J^qy^qr^i^ K = p = L^ip-D (7 = 1 if 2) = 2, 7 = 2 if p >2). We readily verify the recursion formula Ft ^ QFU - KFt, (mod p), since Fi = 1, F2 = Q. In particular, Fz^J-K, F,^Q(Fz^-KJ^'). Now TTs = F3, TTi = FilQ. Hence T3^ J - K, Ti ^ Jp - Rp - KJp-^ (mod p). The first of these results was discussed above. Next, for p = 2, TTi is the irreducible quadratic form q- — P — Iq, so that all quartic forms irreducible modulo 2 are equivalent. For p > 2, 7r4 vanishes for K = pJ, where p^ = 1 — p (mod p). Except for p = §, p is a quadratic Galois imaginary since P^' ^ 1 — pp = p (mod p). Thus 7r4 is a product of J — 2iv and ^{p — 1) irreducible quad- ratic forms in J, K. Some of the latter yield a quartic in q and I which is irreducible; others yield a quartic which is a product of two irreducible quadratics modulo p. A simple discussion shows 64 THE MADISON COLLOQUIUM. that the number of irreducible factors of TTiiq, l) is 6^' + ^ + 1 if p = Sk + t (t = ± I or - 3), but is 6k + 2 'd p = 8k -}- 3. We have therefore the number / of genera of irreducible quartics modulo p. For quintics and septics, the analogous discussion is simple, for sextics laborious. We may utilize similarly the invariants (16) of the group on m variables, obtain expressions in terms of them of the product of all forms in m variables of specified types (as quadratic forms transformable into an irreducible binary form, non-vanishing ternary forms, non-degenerate ternary quadratic forms, etc.), and hence draw conclusions as to the equivalence of forms of the specified type.* * Transactions of the American Mathematical Society, vol. 12 (1911), pp. 92-98. I,ECTURE IV MODULAR GEOMETRY AND COVARIANTIVE THEORY OF A QUADRATIC FORM IN m VARIABLES MODULO 2 1. Introduction. — The modular form that has been most used in geometry and the theory of functions is the quadratic form (1) qm (x) = ^CijXiXj + ^hiX^ {i,j= 1, •••,m; i < j) with integral coefficients taken modulo 2. In accord with Lecture III, we shall use the term point to denote a set of m ordered elements, not all zero, of the infinite field Fi composed of the roots of all congruences modulo 2 with integral coefficients. We shall identify such a point {xi, • • • , Xm) with {pxi, • • • , pXm) where p is any element not zero in F^. The point is called real if the ratios of the a;'s are congruent to integers modulo 2. Let the Cij and 6, in (1) be elements not all zero of the field F^. Then the aggregate of the points (.r) = (.Ti, • • • , Xn^ for which qjn{x) = (mod 2) shall be called a quadric locus, in particular, a conic if to = 3. The locus is thus composed of an infinitude of points, a finite number of which are real. While our results are purely arithmetical, we shall find that the employment of the terminology and methods of analytic projective geometry is of great help in the investigation. Usually the proofs are given initially in an essentially arithmetical form. In case a preliminary argument is based upon geometrical intuition, a purely algebraic proof is given later. The geometry brings out naturally the existence of a linear covariant, which is important in the problem of the determination of a fundamental system of covariants. 2. The Polar Locus. — The point (/c?/i + Xzi, • • • , Kijm + Xz^) is on q{x) = if (2) K\{y) + K\P{y, z) + Vg(z) = (mod 2), 6 65 66 THE MADISON COLLOQUIUM. ' where (3) Piy, z) = ^Cij{yiZj + yjZ^) {i, j = 1, '•■,m;i< j). If (y) is a fixed 'point, all points (s) for which P{y, z) = (mod 2) are said to form the polar* locus of (y). For (z) = (y), each summand in (3) is congruent to zero modulo 2. Hence the polar of (y) passes through (y). If (2) is on the polar of (y), (2) has a double root k : X and the line joining (y) and (2) is tangent to ^ = 0. We may write (3) in the form (3') P(y, z) = uiyi + • • • + w^i/m, where ■Ml = C12Z2 + C13Z3 + C1424 + • • • + CimZm, U2 = CnZl + C2323 + C24Z4 + • • • + C-Zm^, ^^^ 7/3 = CuZi + C2322 + C34Z4 + • ■ • + CsmZm, 1(m = CimZi + C2m22 + C3mZ3 + ' ' * + Cm-lm^-l- There is a striking difference between the cases m odd and m even. 3. Odd Number of Variables; Apex; Linear Tangential Equation. Let m be odd. Then the determinant of the coefiBcients in (4) is congruent modulo 2 to a skew-symmetric determinant of odd order and hence is identically congruent to zero. Hence we can find values of Zi, • • • , Zm not all congruent to zero such that Ui, • • • ,Um. are all zero modulo 2. Thus the polars of all points (y) have at least one point in common. We shall limit attention to the case in which the pfaffians (5) Ci=[23.--m], C2=[134---m], •••, C„,= [12 • • • 7/i-l] are not all congruent to zero. The point (Ci, • • •, Cm) shall be * Take *c = 1 and let (2) be a point not on q{x) = 0. Then (2) is a quad- ratic congruence in X with coefficients in Ft and hence has two roots Xi and X2 in that field. Now the points {y) and (2) are separated harmonically by {y + X12) and (y + X22) if and only if Xi s — X2, that is, if Xi = \z (mod 2). But the condition for a double root of (2) is P = (mod 2). INVARIANTS AND NUMBER THEORY. 67 called the apex* of the locus q(x) = 0. Now each Ui ^ if Zi — Ci, • ' ' , Zm^ Cm. Hence, for m odd, the polars of all points pass through the apex. If {y) is any point not the apex, the line joining {y) to the apex is tangent to q{x) = (§ 2). Thus any line through the apex is tangent to q{x) = 0. For TO = 3, it is true conversely that, if the line (6) XuiXi = (mod 2) is tangent to q{x) = 0, it passes through the apex, so that (7) K = XC^^i is zero modulo 2. Taking, for example, Uz =1= 0, we obtain by eliminating Xz from (6) and q(x) = a quadratic equation in Xi and X2 whose left member is the square of a linear function modulo 2 if and only if the coefficient of XiXi is congruent to zero. But this coefficient is the product of /c by a power of Uz. Thus K = is the tangential equation of q{x) = 0. The last result is true for any odd m. The spread (6) is said to be tangent to q{x) = if the locus of their intersections is degenerate. Taking «,„ 4= 0, and eliminating Xm, between (6) and q{x) = 0, we obtain a quadratic form whose discriminant, defined by (24), equals a product of k by a power of Um, and hence is degenerate if and only if k ^ 0. We thus have geometrical evidence that k is a formal contra- variant of q(x), i. e., an invariant of q(x) and HuiXi. To give an algebraic proof, note that k is unaltered when xi and Xj are interchanged, while (8) Xi = Xi + Xi, X2 = X2', • • •, Xm= xJ replaces q{x) by q'(x') in which the altered coefficients are (9) h'z' = b2 + hi -{- cn, Cu = Coi + cu (i = 3, - ■ • , m). * After these lectures were delivered, I learned that Professor U. G. Mitchell had obtained, independently of me, the notion apex (" outside point ") for the case m = 3, Princeton dissertation, 1910, printed privately, 1913. 68 THE M.VDISON COLLOQUIUM. The pfaffians Co, • • • , Cm are unaltered modulo 2, while (10) Ci'=Ci+C2, W2'=W2+Wi, Ui'=Ui {i^2) (mod 2). Hence k is unaltered modulo 2. Note that (11) ^2 = C\J. IX. Cl2 Cl3 ■ • Cim Ui Cl2 • • C23 • • • ' Com • • • U2 • Clm C2m C3m • •• Wm Wl W2 Us •• llm (mod 2). "We saw that Ci, - • • , Cm are cogredient with Xi, ' ", ^m- This is evident from the fact that the apex is covariantively related to q{x). Hence if we substitute Ci for Xi, - > • , Cm for Xm in (1), we obtain the formal invariant (12) qUC) = i:CijCiCj + i:biC^ (i, j = l, • . • , m; i < j). If this invariant vanishes, the apex is on the locus, which is then a cone. Indeed, by (2), every point on the line joining (C) to a point on q{x) = lies on the latter. Hence q{;x) can be transformed into a form in 7?i — 1 variables and hence has the discriminant zero. To argue algebraically, let new variables be chosen so that the apex becomes (0, • • • , 0, 1). The polar of any point (?/) passes through the apex. Taking Si = 0, • • • , Zm-i = 0, Em = 1 in (4), we see that the polar (3') becomes CimVi + • • • + Cm-\mym-\y which must vanish for arbitrar}^ y's. Hence hmXm is the only term of (1) involving Xm. But the apex is on the locus. Hence 6m = and q{x) is free of Xm. The converse is obvious from (5). Whether m is odd or even, q{x) has the invariant (13) Am = n(co- + 1) (i, j = I, ••',m;i< j). This is evidently true by (9) or as follows. If Am = 1 (mod 2), every Ca = and q = (2fe,.T,)-; while if Am = 0, at least one Cij is not congruent to zero, and q is not a double line. Hence the product Amq(x) is a co variant; in fact, the square INVARIANTS AND NUMBER THEORY. 69 of the linear eovariant Am^biXi. We shall see however that there exists a more fundamental linear eovariant. 4. Covariant Line of a Conic. — Since we shall later treat in detail the case m = 3, we shall replace (1) by the simpler notation (14) F(x) = aiX2X3 + a2XiX3 + 03^:12:2 + biXi- + biXi' + 632-32. Its apex is (ai, 02, as). Its discriminant (12) is (15) A = F{ai, 02, as) = aia^as + a-^bi + a^^^ + az^bz. The invariant (13) becomes (16) A = aia^az {(Xi = a,- + 1). Consider a form (14) with integral coefficients and not the square of a linear function. Then not every Oi is congruent to zero modulo 2. By an interchange of variables we may set as = 1. Replace Xi by Xi + aiXz and X2 by X2 + 02^:3. We get Z1Z2 + 6iA^2 + 62X2^ + Axz\ Let A = 1. Replace xz by Xz + biXi + 60X2. We get (17) = Z1Z2 + AV. The only real points on <^ = (mod 2) are (1, 1, 1), (1, 0, 0), (0, 1, 0). In addition to these and the apex (0, 0, 1), the only real points in the plane are (1, 1, 0), (0, 1, 1), (1, 0, 1). These lie on the straight line (18) Xi + Z2 + Xs = (mod 2), Hence with every non-degenerate conic modulo 2 is associated covariantly a straight line. The inverse of the transformation used above is Xi = a!i + aiXz, X2 = a;2 + a2Xz, Xz = biXi + 62^:2 + (1 + oi&i + a2b2)xz. It must therefore replace by the general form (14) having 70 THE iLU)ISON COLLOQUIIBI. «3 = A = 1. It actually replaces (18) by (6i + l).Ti + (62 + l).'r2 + (63 + aiao + 1).T3, in which we have added A + 1 = to the initial coefficient of Xz. Guided by symmetry, we restore terms which become zero for 03 = 1 and get ,,,, X = Z (/3.- + 1K-, (19) .=1 i3i = 61 + ociOis, i32 = 62 + aitts, /^s = &3 + (Xia2. Making the terms homogeneous we obtain the formal co- variant (20) L = Bixx 4- 52.1-2 + Bzxz, Bi = bi^ + a2Ct3 + a2^ + as', -B2 = bo~ + ctiOs + ai^ + ai^, (21) -S3 = &3^ + 01^2 + ai^ + a2^. Under the substitution (aiaj)(bibj) induced upon the coefficients of F by (xiXj), we see that Bi and Bj are interchanged. Under (9), viz., (22) bo' = 62 + &i + 03, a/ = ai + a2 (mod 2), there results (23) 5i' = Bu B2' ^ B2 + Bu Bs' ^ Bs (mod 2). Hence (20) is a formal covariant of F. For other interpretations of L see § 8. 5. Even Number of Variables. — The determinant of the coef- ficients in (4) is congruent modulo 2 to the square of the pfaffian (24) A^ = [123 • • • m]. This is in fact the discriminant of q-m, which is degenerate if and only if A;„ = (mod 2). I have elsewhere* discussed at length the invariants of qm- * Transactions of the American Mathematical Society, vol. 8 (1907), p. 213 (case m =2); vol. 10 (1909), pp. 133-149; American Journal of Mathematics, vol. 30 (1908), p. 263; Proceedings of the London Mathematical Society, (2), vol. 5 (1907), p. 301. INVAEIANTS AXD NUMBER THEORY. 71 If Am ^ (mod 2), we can solve equations (4) for the zs. Substituting the resulting values into q{z), we obtain the tangen- tial equation Um = of q(x) = 0. For m = 2 and m = 4, we get U2 = C12U1U2 + biiii^ + 6l2^2^ (25) U4. = [l234]2C34?'rW2+2(C23C24C34+&2cL+&3cL+&4C23)Wl^- Bordering the algebraic discriminant of (1), we find that (26) 2U^ ^ 26i C12 Cl3 • • ' Clm Ui C12 2bo C23 • • Com Uo Clm Clm Czm ' •• 2hm Um Ui 111 II3 •• Um (mod 4). Finally, let A^ ^ (mod 2). Then all of the first minors of the matrix of the coefficients in (4) are zero modulo 2. Hence the polars of all points have in common the points of a straight line S. Since its discriminant vanishes, q{x) can be transformed linearly into a quadratic form in a:i, • • • , Xm-i, which therefore represents a cone with the vertex (0, • • •, 0, 1). Let (2) be the vertex of the initial cone q{x) = 0. If (x) is any point on the cone, (x + \z) is on the cone, and, by (2), P{x, z) is congruent to zero identically' in .ti, •••, Xm- Hence the linear functions (4) all vanish. Thus the line S meets the cone in its vertex, and Zm^ is the discriminant of qm~i{x), while Zi^ is obtained from that discriminant by interchanging m and ^. For example, if m=4, 24^ = C12C13C23 + &1C23 + &2C13 + &3C12, • • • , 2r = C23C24C34 + boch + 63C24 + 64023- The product of the general form (1) by 5 = A;„ + 1 is a quad- ratic form whose discriminant is zero modulo 2 and hence has the vertex (dzi, •••, dzm), where Zi^ has the value just given. Hence 8zi^, • • • , dzm^ are cogredient with Xi, • • • , Xm- 6. Covariant Plane of a Degenerate Quadric Surface. — The product of g4 by 5 = [1234] + 1 is a quaternary form / whose 72 THE MADISON COLLOQUIUM. discriminant is zero and hence can be transformed into a form (14) free of x^. With this cone F = is associated covariantively the plane / = 0, where I is the ternary covariant (19). Hence/ has a hnear covariant L which reduces to I when 64 = 0, Cu = (i = 1, 2, 3). Relying upon symmetry and the presence of the factor d, we are led to conjecture that L = 5{6i + 1 + (Ci2 + l)(Ci3 + l)(Cu + 1)1.^1 + • • • ^^'^^ + 5 { 64 + 1 + (Ci4 + 1) (C24 + 1) (C34 + 1) } X,. It is readily verified algebraically that X is a covariant of 94. There is a simple interpretation of L. If [1234] ^ (mod 2), then 5=0 and L is identically zero. If [1234] = 0, q^ is de- generate and can be transformed into (f) = a'l.ro + 0*3^ or a form involving only xi and X2. In the former case, L = xi-\- X2 -{- Xz. Of the 15 real points in space, the seven (100.t), (OIO.t), (lll.r) and (0001) are on the cone 0^0, the two (OOLr) are on the invariant line S through the vertex (0001) of the cone and the apex (0010) of the conic cut out by 0*4 = 0, while the remaining six (101a:), (011a:), (110a:) lie on the plane L = 0. Hence with a degenerate quadric surface, not a pair of planes, is associated covariantively a plane, just as a line (19) is associated with a non-degenerate conic (14). Every linear covariant is of the form IL, where I is an in- variant. Every quadratic covariant is a linear combination of the lU and Iq^. 7. A Configuration Defined by the Quinary Surface. — A q^ whose discriminant is not zero modulo 2 can be transformed into F = a:ia:2 + a:3a:4 + a^-. The 15 real points on i^ = (mod 2) are given in the last column of the table below. In addition to these and the apex (00001) of F, there are just 15 real points in space: 1 = (00011), 2= (01001), 3= (01011), 4= (00101), 5= (01101), 6= (00110), 7= (OHIO), 8= (10001), 9= (10011), a= (10101), 6= (10110), c= (11000), cZ= (11010), e= (11100), /= (11111). INVARIANTS AND NUMBER THEORY. '3 These lie by threes in exactly 20 straight lines, which occur in the columns of the table, with the heading " Sides." With these lines we can form exactly 15 complete quadrilaterals, the three diagonals of each of which intersect* in a point on F ^ 0, given in the last column. The columns, with the heading "Plane," give the equations defining the plane of the quadrilateral. In each case, the two equations of the plane have in common with F ^ a single real point, the intersection of the diagonals. Thus the real points on i^ = are its points of contact with these tangent planes. Sides Diagonals Plane Inter- section 146 157 356 347 13 45 67 xi=0, X3+Xi-\-Xi = Q 01000 146 lab 496 69a 19 4a 66 X-i=Q, X3+X4 + X5=0 10000 146 W 4d/ 6de Id 4e 6/ Xi=Xi=X3-\-Xi+Xi 11001 157 lab 5ac 76c Ic 56 (« Xi-\-X2+Xz=Xz-\-Xi-\-Xi=0 11011 157 lef 58e 78/ 18 5/ 7e X2=X3, Xi=Xi+Xi-\rXt 10010 lab lef 2ae 26/ 12 af 6e Xi=X3, X2=Xz-\-Xi + Xi 01010 28c 29d BSd 39c 23 89 cd X3=0, Xi=X2-\-Xi 00010 28c 2ae 5ac 58e 25 8a ce X4 = 0, Xi=X2+Xi, 00100 28c 26/ 78/ 76c 27 86 cf X3=Xi, Xl=X2 + Xi-\-X5 00111 29d 2ae 69a 6de 26 9e ad Xi=X3+Xi=X2+Xi 01111 29d 26/ 496 4d/ 24 9/ bd Xi=Xi, X2=X3+Xi+Xi 01100 347 38d 4d/ 78/ 3/ 48 Id X2=X4, Xi=X3+Xi+Xi 10100 347 39c 496 76c 36 4c 79 Xi=Xi+X2=X3+X5 11101 356 38d 58e &de 3e 5d 68 X2 = X3 +Xi, Xi = X2 +Xs 10111 356 39c 5ac 69a 3a 59 6c Xi,=Xi+X2 = Xz + Xi 11110 8. Certain Formal and Modular Covariants of a Conic. — For conic (14), the polar form is (28) Hence if two sets of variables yi and 2;^ be transformed cogredi- ently with the set Xi, this polar form (28) is a covariant of F and the two points (y), (s), in an extended sense of the term * The dual of the theorem of Veblen and Bussey, " Finite projective ge- ometries," Transactions of the American Mathematical Society, vol. 7 (1906), p. 245. «1 ^2 az 2/1 2/2 2/3 Sl 22 23 74 THE M.\DISOX COLLOQUIUM. covariant. In particular, if we take {y) = {x), (z) = {p^), we obtain a covariant of F in the narrow sense used in these lectures. In particular, (29) K = Xi 02 Xi fl3 .^•3 Xl- X2^ .T3^ M = ai a2 as Xl X2 Xs Xl' Xo' xz' are formal covariants of F. While the discriminant A, given by (15), is a formal invariant, (16) is not. But (30) (31) ^ + A + 1 = q; (mod 2), a being a formal invariant of F. By (23), the B's are contra- gredient to the x's and hence to the a's, so that (32) Ai = XaiBi = Xuihi^ + "Eaia/ + aiaaaa is a formal invariant. For integral values of ai, hi, (33) Ai ^ A = 2ai(/3v + 1) (mod 2). Any form with undetermined integral coefficients Ci, C2, • • • , taken modulo 2, has, by (21) of Lecture I, the invariant (ci + 1)(C2 + 1) • • ■• Thus (16) is an invariant of (7) and hence of F. Likewise from (19) and F itself, we obtain the invariants (34) J = /3i/52i33, AJ = AYl{hi + 1). In (6) we made use geometrically of (35) X = W1.T1 + U2X2 + W3.T3. Now F + ^X^ is congruent modulo 2 to the quadratic form derived from F by replacing each hi by &,- + tu^. Making this replace- ment in A, we see that the coefficient of t is congruent to k^, where (36) K = aitli + 02^/2 + 03^3 is therefore a formal invariant* of F and X. Making the same * Since (36) is a contravariant of F, Zai(dC/dXi) is a covariant of F if C is. Taking Q2, Qi, L as C, we get K, M, A, respectively. INVARIANTS AND NUMBER THEORY. 75 replacement in J and taking t and ui to be integers, we obtain as the coefficient of i ^ ^^ (37) + PlUlUz + PzUlUz + W1M2W3, a modular invariant of F and X. By the theorem used above, (38) U= (2/1+ l)(i/2+ 1)(«3+ 1) is an invariant of X. In w + w + 1, we replace |Si by the con- gruent value -Si + 1, and render the expression homogeneous in the w's and 5's separately. We get (39) CO = 2(5i52 + B{~ + 52^)^3- + ^B^hmz, a formal invariant of F, X. For, it is unaltered by the sub- stitution induced by {xiXj), and by the substitution (23) and (10) induced by (8). Let the coefficients of F be integers not all even. Then (39) becomes (39') 2(^1/32 + 1)W3- + 2(i3i + 1)W2W3. Its covariant L is identically zero. Hence, hy the table in § 9, if CO is not identically zero it can be transformed into u^ -\- u-r + W1W2 and hence vanishes for a single set of integral values of Ml, U2, Uz. These are seen to be Ui = /3i + 1. Hence* the line L = is the only line with integral line coordinates on the line locals (39). The invariant A for (39) is J (its discriminant is zero, as just seen). Thus a knowledge of any one of the concomitants L, J, CO implies that of the other two. The covariance of K in (29) implies that (40) ^1 = X2 Xz X^ X^ ^2 = Xi Xz Xi^ Xz- •Tl Xi ^3 = Xi' X2^ * Also thus: just as the point conic F = determines its hne equation (36) and hence its apex (a), so the covariant line conic (39) determines the point equation 'LBC-Xi = 0, which is the line L = for integral values of the coef- ficients. 76 THE MADISON COLLOQUIUM. are contragredient with ai, a^, cts and hence with Xi, X2, X3, and therefore cogredient with ? 1, mFL'^~'^ has this same leader. For co = 1, C = ???(63a:ic»;2a-3 + h\(X2(XzXi + 62aia3a'2)i which satisfies (61) only when m = 0. Hence every linear covariant is a linear function of L, AL, AL; every covariant of odd order co > 1 differs from a linear combination of L", AL"^, AL", FL'^~^ hy a covariant ichose leader is zero. 12. In the covariants of order 47i (64) /Q2", IF-"", U\ F-^-^U (/ an invariant), the coefficients of Xz""^ are respectively /, hzl, bz + aiao + 1, bz(Xia2. Linear combinations of these give every leader (59). Hence every covariant of order 4n differs from a linear combination of the covariants (64) by a covariant whose leader is zero. 13. In the covariants of order co = 4/i + 2 (65) IQ2''F, Q2"/-'-, AQs^i^ {I an invariant), the coefficients of 0:3" are respectively bzL bz + aia2 +1, A + 63(A + aia2az). The sum of the third function and bz(A + A) is A + bzaia2' Hence any covariant C is of the form P + C, where P is a linear IXVARIANTS AND NUMBER THEORY. 81 combination of the co variants (65), while C is a covariant whose leader is an invariant. For co = 2, C = SX3^ + S1.r3.r1 + &i2 + X2({). This is transformed by (51) into a function having Si as the coeflficient of Xi'. Since S is an invariant, *Si = S. Thus every coefficient of C equals 2. If 5/ is the function obtained from iS,- by the substitution (50), then (67) 5'/ = Si, S2' = S-\-Si^- S2. Now Si is unaltered also by the substitutions (22) and (68) as' = as + Oo, h.' = bo + 63 + 01 (mod 2), induced on the coefficients of F by the transformations (8) and (69) .Ti = .Ti', .To = Xo', X3 = X3' + xV. 15. A fundamental system of invariants of F, under the group V generated by the transformations (8), (51) and (69), is given by A, A, J, 02, 63, OiCk:2 and (70) iS = 61(63 + a2). It suffices to prove that these seven functions, which are evidently' invariant under F, completely characterize the classes of forms F under F. There are six cases. (i) 63 = a2 = 1- Replacing xx by .Ti + ai.T2 and .T3 by X3 + azX2, we get F = fixi' + A.r2' + .T3' + .ri.r3. (ii) 63 = 1, 00 = 0, flia:2 = 1. Replacing Xs by .1-3 + Os.i'i, we get F = Aa-i2 + 62.1-2- + .rs- + x,xs. 7 82 THE MADISON COLLOQUIUM. If A = 0, then h%^ J. If A = 1, we replace xi by Xi + 62-^2 and get a^i^ + .^3^ + .^2.^3. (iii) &3 = 1, a2 = CLiOLi ^ 0. Replacing .1-3 by Xz + hiXx + hiX2, we get Xz' + Aa:i.T2. (iv) 63 = 0, a2 = 1. After replacing .T3 by .^3 + 03.7:2, we obtain a form with also 03 — 0. Taking this as F, and replacing a'l by .Ti + aiX2, we get hix-^ + Aa:2^ + 2:1.^3. Replacing xz by 0-3 + hiXi, we get A.T2^ + XiXz. (v) hz = 02 = 0, 010:2 = 1. Replacing .T3 by .T3 + 03a;i + 62-^2, we get ^x^ + 0:23:3. (vi) 63 = 02 = 010:2 = 0. Then F is the binary form/ in (46). The effective part of F is now the subgroup Ti generated by (8). Now /3=&i, ^+1 = 03, J = 5 + (61 + 1)03, 5 = 62(6i + «3). These seminvariants hi, az, -B of / completely characterize the classes of forms/ under Fi. For, if 03 = hi, f = hixi^ + Bx2^ + 6i.Tia:2; while if 03 = 5i + 1, we replace .Ti by Xi + 620*2 and get hiXi^ + {hi + l).Ti.'r2. 16. The number of classes of forms F in the respective cases (i)-(vi) is 4, 3, 2, 2, 6. Hence there are exactly 19 linearly independent invariants of F under the group F. As these we may take 1, 02, Oia2, ^> hz, hztto, hzaia2, hzA, A = 6101 + • • •, 02A = &iflia2 + • • •, (71) /3 = 61(63 + 02), a.^ = 6163O0, INVARIANTS AND NUMBER THEORY. 83 (71) Afi=bi(bz+1)A, b3A=bib3ai-\ , a2bzA = bib3aiao-i , J = bxbibz + • • •, a^J = bib^bza-i + • • •, bzJ = bibibziaiQi + oi + 02) + • • •, AJ = bib2bzA + • • • . These are linearly independent since the first eight do not involve 61, while all the terms with the factor bi in the next seven are given explicitly, likewise all with the factor 6162&3 in the last four. Hence the 19 functions (71) form a complete set of linearly independent invariants of F under the group T. 17. Hence, in § 14, Si is a linear combination of the functions (71). By (670), S + Si is of the form (57) if *S2 be denoted by (56). Now 0361 occurs in J, AJ, bzJ, aiJ, AjS, but in no further function (71). In the first three, azbi is multipHed by the linearly independent functions (62), respectively; in the last two by bzaia2 and aiazibz +1), whose sum is congruent to the first function (62). Hence the part of *S + *Si involving J, • • • , Aj3 is a linear combination of (72) (63 -\- 02) J = bibobzaiao + bibzaiaoocz, (73) J+bzJ + A^= (63 + l)(6i6oa:a2 + M + A). But 61 occurs in just six of the functions (71) other than the five just considered. Thus the factor pai of 61 in (57) is a linear combination of the coefficients of bi in (72), (73), /3, 02/3, A, aoA, bzA, a'UbzA. Now ai is a factor of the coefficients of 61 in all except the second; third and fourth, while in these the coefficients are (63 + I)b2aia2, 63 + «2 + 1, a2&3 and are linearly independent. Hence (73), jS, a2/3 do not occur in *S + Si. By (57), the latter has no constant term and hence involves 1, A only in the combination ^ + 1. This cannot occur since the total coeSicient of az must be of the form jjR and hence vanish for 63 = 02. At the same time we see that the sum of the constant multipliers of A, a2A, 63A, a263A is zero modulo 2. Hence *S + Si is a linear combination of the functions 84 THE MADISON COLLOQUIUM. «2, bs, hza2, a\at, and the last six in (74) below. Like (57), this combination must vanish for a\ = 0, 63 = a^. Since all but the first three of the ten functions then vanish, the sum of the multipliers of these three must be zero modulo 2. Hence S -{- S\ is a linear combination of &3 + «2, aiihz + 1), a\0i2, ^aOiaa, hzA, ^'^^ Aa„ A(63+l), A(ao63+l), (bs + a^W. 18. Without altering the invariant S, we may simplify Si by subtracting from C constant multiples of Z"*""^ K and its product by A, where K is given by (29), and hence delete a2{b$ + 1) and A(ao63 + 1) from the terms (74) of Si. Then Si = S + mAa-i + miA{b3 + 1) + vi2{b3 + a2)J + mzibs + f/o) + m^aia-i + mjjsaia-z + m^bzA. The coefficient T of a:3"~\i-2 in C is obtained from Si by applying the substitution (0102) (6162) induced by (3:1X2). In view of the transformation (8), we see that T' = T -\- Si, where T' is derived from r by (22). Hence S = (m + mi)A + mib^A + viobsJ + (niA + m5&3)(aia2 + ai + 02) + W3&3 + mobs A. Let S be the sum of the second member and the function ob- tained from it by the substitution (0203) (&2^3)- Thus S = 0. Taking 63 = b^, we get W4 = W5 = 0. Then 2 = (62 + ^3)^, / = '"lA + niiJ + ??i3 + msA. Applying to S the substitution (68), we get (62 + ai)7 = 0, Applying (aia3)(6i63) to the latter, we get (62 + ci^)! = 0. Adding, we get (ai + ch)! = 0. Applying (50), we see that asl = 0. Then each aj = 0, so that I = gA, where gf is a constant. By 2 = 0, 2 differs from a linear combination of the co- variants (65) and Qo'^~^E by a covariant whose leader is zero. 19. Regular and Irregular Covariants; Rank. — A covariant shall be called regular or irregular according as it has not or has the factor Lz, given by (43). The quotient of an irregular covariant by L3 is a covariant. Hence the determination of all irregular covariants reduces to that of the regular covariants. If a covariant has a linear factor it has as a factor each of the seven ternary linear functions incongruent modulo 2, whose product is Lz. Hence a regular covariant has a non- vanishing component involving only xi, .T3. In a regular covariant C without terms .Tj" (i. e., with leader zero), this component has the factors Xi, Xz and (by the covariant property) also .Ti + Xz. The product of these three linear factors was denoted by ^2 in (40), Let ^2"* be the highest power of ^2 which is a factor of the component and let n be the degree of the quotient in the .r's. Then C may be given the notation 3 (76) Rm,n= ^fi^t^ + XiX2Xz4), i=l where, if n = 0, /2 is a function of the a's and 6's not identically zero, while, if n > 0, /2 is a function also of Xi, Xz in which the coefficients of .ri" and .T3" are not zero; /i is a function of x^, Xz; fz of Xi, X2. The regular covariant (76) shall be said to be of rank m. In an irregular covariant the component free of X2 is zero and hence is divisible bj^ an arbitrary power of ^2 ; it is proper and convenient to say that an irregular covariant is of infinite rank. Any covariant of rank zero differs from one of rank greater than zero by a polynomial in the known covariants (77) A, A, J, F, L, Q,. This is a consequence of the theorems in §§ 11-18, where the polynomial is given explicitly. Any product, of order co in the 86 THE MADISON COLLOQUIUM. x's, of powers of the covariants (77) can be reduced by means of the syzygies JL = 0, AD = AF, (A + ^ + / + 1)(FL + 70 = 0, (78) AK = 0, FU + (^ + A)L' + AF + AQ2 = LK, F' + Q,F = UK + (A + JW + (A + l)LG + (.4 + l)Qi, to a sum of covariants of order co given in §§ 11-18 and a linear function, with covariant coefficients, of K, Qi and G= Q2L+D = 2t2[^3(^i + 1).^/ + (/3i|33 + D.ts.ti (79) + ^1(^3 + Dxi"] + xxX2X^m + /32 + /33 + 1) X (a;i.T2 + a^i.Ts + X2X3) + S(/3i + l)a.'r]. Here G and X, given by (42), are of rank 1, while Qi= ^2^+iK2 ( ) is of rank 2. As this theorem is not presupposed in what follows, its proof is omitted. However, it led naturally to the important relations (75) and (79) and showed that no new combinations of the covariants (77) of rank zero yield covariants of rank > 0, a fact used as a guide in the investigation of the latter covariants. Regular Covariants R,n.o, §§ 20-22 20. A separate treatment is necessary for covariants (76) with w = 0. Then each fi is a function of the coefficients ay, bj. Since the factor ^3*" of the part /s^s"* of jR^o free of X3 is unaltered by every linear transformation on Xi and X2, /a is a linear com- bination of the functions (49) and their products by 63. Also, fs must be unaltered by (80) Xi = xi + X3: ai = ai + a^, 63' = 63 + &i + a?. Both conditions are evidently satisfied by the ternary invariants and by a^ and q, in (47). In view of (53), we may employ AJ, J, ttsA, A, asJ, qA, A to replace in turn bzaia-ia^q, b-iOiia-iaa, azj, j, a-Jj^q, aiazci^q, aia-zas, INVARIANTS AND NUMBER THEORY. 87 since a term previously replaced is not introduced later. Thus /s is a linear combination of these seven functions, as, q, azQ, and ai(X2, Ciiaiq, hz, 63O3, hzq, hzaia-i, hzocia^q, bzj, bzazj. Give to any linear function miaicxi + • • • of these the notation (T = aaibz + |Sai + 763 + 5. Call e the increment 61 + ^2 to 63 in (80) and employ e to eliminate 61. Then a is unaltered by (80) if and only if ae = 0, aaz = 0, fiaz = ye (mod 2). Since 63 does not occur in q or j, nor ai in q, we have a = m&a2 + 7n7a2q + ms{e + 02 + az) + ^9^3(6 + 02 + as). Thus ae = gives 711^ = m? = 0, mg = m^. Then aaa = gives TO9 = 0. Now j3 = mia:2 + W2a2?, 7 = Tiiz + m^az + Wsg, and jSaa = 7^ readily gives o- = 0. ^wy function of 63 anc? ^Ae invariants (49) 0/ / anc? /, ?f ^icA Z5 unaltered by (80) , Z5 a linear combination of the ternary invariants (45) and az, q, azq, asA, azJ, qA. 21. For n = and w even, there exists a co variant (76) in which fz is any function specified in the preceding theorem. For, if / is any ternary invariant, IQi^^^ has fz = /. By (42) and (41), K"" and W^''- are of the form (76) with fz = az and ^1^2 + 1> respectively; they may be multiplied by any invariant. By (19) and (47), we have (81) i8iiS2 + 1 = q + azA + A + 1, azq = azA + qA + OzJ. Hence we obtain q, then qA, qA, and therefore azq. Any co- variant with n = 0, m even, differs by an irregular covariant from a linear function of IQr^hK^,hW^'' {I=l,A,A,J,AJ;Ii=l,A,J;h=l,A,A). 88 THE MADISON COLLOQUIUM. 22. For n= and m odd, we may delete the terms cts/i from fz by use of IiK"^. First, let m = 1 and apply transformation (51) ; we get R'= U,'+M2 + (/l + /3)s^3' + {X^'X^'XZ' + X^'\'). Thus ^ = 0. Since fz=l-\- hf], condition /i + /a = /s' gives I = hittibi + a2&2 + 0^363 + aoaa + a 102). Add to this the relation obtained by permuting the subscripts 1, 2. Thus = hibi + &2 + 020:3 + aiaz). The increment under (22) is 72(61 + 03 + 0203) = 0. Now I2 is of the form x -\- yA -\- zA, where x, y, z are constants. From the terms in 61&2, we get y = Q. Then x = z = 0. The only covariants are therefore IiK. Second, let m > 1. Then XTF('»-i>/2 jg ^f ^he form (76) with /s = tta? + o-z, by (81 1). Hence we may set fz — I -\- cq-\- dqA (c, d constants). In R given by (76), let g denote the coefficient of (83) X1X2XZ • a-2"'a-32"'-l In the function derived from R by the transformation (51), the term corresponding to (83) has the coeflBcient g + /i, since by (82) the ^i parts contribute only one such term, that from fi^r'%'- Now fi= I + cq' + dq'A, q' = h.hz + (62 + hz)ai. When g is given the notation (56), g' — g = f\ is the function (57). But azhi occurs in/i only in J and AJ and in them with the linearly independent multipliers (62). Hence I = niiA + 1) + ?i2A. The coefficient of az in /i is now niaia2 + 712(^102 + 63) + dq'aiUo = p(bz + 02). INVAEIANTS AND NUMBER THEORY. 89 Taking &3 = 02, we see that ni ^ n2 = d ^ 0. Thus /i = cq'. By (57) for ai = 0, 63 — 02, we get c = 0. Any covariant icith n = and m odd differs by an irregular covariant from a linear junction of K"", AK*^, JK"* and, if m > 1, KW^"^-^^'-. CovARiANTS OF Rank Unity, §§ 23-26 23. Henceforth let m > 0, n> in (76) and set (84) /2 = Sxs- + Six,--'xi + S2X,--W + • • • (S + 0). Since S is unaltered by the group F of § 15, it is a linear com- bination of the functions (71). We may omit the functions 02(63 + 1) and Aa2(63 +1), since Z'"i'^ is of the form (76) with S = 02(63 + 1). Thus (85) 5 = 7+ 02/1 + 63/2 + kia^a^ + hh^aia^+h^+h^aifi+hA^, where I is any invariant, 7i a linear function of 1, A, J; /a one of 1, A, A, J; while ^3 = 61(63 + 0:2). First, let m = 1. If T and B are the coefficients of 0*2" in fz and /i, transformation (51) replaces the covariant (76) by a function in which, by (82), the coefficient of Xix^'"^^^ is (86) T+B= T', where T' is derived from T by the induced substitution (50). But T is obtained from 1 by a linear function of KL^, AKL"^. 24. For m = 1, n = Av, we may delete aoli from (85) by use of IiKQ2\ Set /i = 5a-2" + h ^nXs". Then (51) replaces (76) by R' = USXz" + SiXz^-'Xi + (Si + S2)Xz--W +•••] + ^3/3 + (^1 + ^3)[Bn{Xz- + XZ--W + • • •) + Bn-lXiiXz^-"^ + Xz^-^Xi H )] + (.Tl.T2.T3 + Xi^X^W. Since Si is the increment of ^2, it is a linear combination of the functions (74). By use of L^~^Qi, L'^~^K^ and their products by A and A, we may, without disturbing S, delete from Si 63+q:iq;2+1, Abz, A+b^A+bzaiai, 02(^3+1), 02A(63+1). Hence we may set 'Si = tiibz + tti) + t-ibzaiOi2 + ^3Aa2 + ^4(63 + a2)J. Applying (0102) (6160) to S and Si, we obtain Bn and Bn-i. Let I be the coefficient of .Tj-Ts""^ in 0. By the coefficient of INVARIANTS AND NUMBER THEORY. 91 .Tia-2a-3 • a-2.r3"~^ in R', we have Bn^-Bn-i + l^V. For I given by (56), Bn + Bn-i is given by (57). By the coef- ficient of ^361, we get ^4 = 0. The coefficient of a^ must vanish for 63 = 02. Hence hoii + {h + ^3)ai«2 + hoiiciih = 0, ki= h^ 0, U = hy S = kobziA + ^ + 1 + aittz + a2?'i). The coefficient of ki equals that of ^23:3"' in GFQ2'-' + AKL^ + AKQi". Any covariant icith m = 1, n = 4v, differs from one with m ^ 2 by a linear function of KL"", AKL"", IKQ^", GFQ/-^ (/ = 1, A, J). 25. For m = 1, n = 4v -\- 2, we may delete 02/1 from S, given by (87), by use of IiQ-^'M. The coefficient of ^2^:3'' in q^'G is d = /33(/3i + 1) = ^ + (61 + l)(aia2 + 63) + b^a^az. The coefficient of ki in ^S equals d + QiA + 02(63 + 1), the final term of which was reached in § 23, and 02A above. The coef- ficients of k-a and k^ in S equal Ad and bibz{ax-\-ai)-\-bz{a2b2+aiaz-\-a2az+a2)=Ad-\-a2{J-\-\)-\-a2{bz-\-V), respectively. Any covariant with m = 1, n = Av -{- 2, differs from one with m ^2 by a linear function of KL"^, AKL"^, IQ^G, hq.^M (I = 1, A, A; h = 1, A, J). For use in § 26, we replace Q2''M by Qi'FK, noting that (88) M = {F-{- U)K and that Q^UK differs from KL^ by a covariant of rank 2. 26. By the last four theorems, any covariant of rank 1 differs from one of rank ^ 2 by CK + DG, where C and D are known covariants of rank zero. Taking as C\ and Di arbitrary func- 92 THE AIADISON COLLOQUIUM. tions of the proper degree in the a;'s, of the generators (77) of covariants of rank zero, I found the syzygies needed to reduce CiK-{- DiG to an expression differing from the above CK + DG by a covariant of rank ^ 2, in which those of rank 2 are linear combinations of K^, KG, G^, W, Qi and the new one V = GF' + AQ.G + (A + ./ + l)Q2FK ^here . + ^^'^" + ^^'^' = ''"'''"' + ' ' ' ' (90) v= 02+ 63(1 + oiaz). The only new syzygies needed for this reduction are LG = QoL' -{- L^= W, FLK = AW + AQi + ( J + IW, (91) (F + i^ + Q2)K = (^ + 1)7.3, (A + 1){FG + KL^ + KQo) + JKQo = ALQi + C0Z3, in which co is an invariant not computed. Proof need not be given of these facts since we presuppose below merely the ex- istence of relation (89) which may be verified independently. Of course, the fact that V is the only new covariant of rank 2 was a guide in the later investigation. CovARL\NTS OF EvEN Raxk lU = 2/i > 0, §§ 27-29 27. First, let n be odd. In the covariant (76) replace X3 by Xz + Xi. In view of (82), we get R' = fi'^'T + /3.^3'" + Uax' + ^^3')'^ + (X,X2XS + Xr^X,)4>'. Using the notation (84) for f^, we have »Si' = Si-\- S in f^'- Thus, as in § 17, *S is a linear combination of the functions (74). Now Qi'^L"' and its products by A and .4 + A are covariants (76) with S given by (63). Using also A^*"i", in which S = 02(63+1), and its product by A, we may set S = ^1(63 + 02) + /i2&3aia2 + /i*3Aa2 + ^^'iibs + ao)J. In a:i.T2.T30, let g be the coefficient of a;ia-2a-3 • Xi-^^'W'^'''^ = {x2W)''x3''-^Xi. INVARIANTS AND NUMBER THEORY. 93 Such a term occurs in neither of the first two parts of R', since they are functions of only two variables. To obtain such a term from the third part of R', we must omit terms with the factor ^3- (and hence .rr) and take {x^x^Y'^ in ^i"*^, so as not to make the degree in .T2 too high. Hence if T be the coefficient of Xz^ in /i, (/'= 0, differs from one of rank > m by a linear comhination of JQi'^i" (/ = 1, A, A), 7v"*i", A7i:«Z" and, if n> I, Qi'^-^L^'-W. 28. For m = 2n > 0, 71 = 4^ > 0, the coefficients of ^2'"a*3" in Ql'^<32^ K"^Q2, Qi'-F-^ Qi-L\ K-^L-, are respectively 1, «2, 63, (33 + 1, a.{h3 +1), d = i83(/33 + 1), aod. These may be multiplied by any invariant. Xow 1^3 + 1 + Oo + 63 = aidi, A(/33 + 1) + (A + .1 + 1)63 + h3a2 + A = hzaicco, rf + .4 + /33 + «2(A + 63) = hi{h3 + «2) = /3, a2cZ + 0263 = «2&1&3 = «2/3, ^'W = ^61(63 + 1) = Afi. Hence we have a covariant (76) in which the coefficient of ^i^Xz* is any linear combination of the functions (71). Hence the 94 THE MADISON COLLOQUIUM. covariant differs from one of rank > m by a linear function of the covariants (92), the 'products of the first three by any invariant except 1, the products of the fourth and fifth by A and the product of the sixth by A. 29. For m = 2/z > 0, n = ^v + 2, the coefficients of ^2"*^3" in (93) MK^^-^q.^ K^L--, GK^^-'Qi", F'"-'Qi^, L^Q^^ are respectively G2, 02(&3+l), azbsibi -j- 1) , bs, 63 + aiaa + 1. Linear combinations of products of these by invariants give* a2, a^A, a^J, a^bz, Aa^bz, 026163, ^63, (iioci, A + 63^10:2. Since S and Si are unaltered by the group F of § 15, they are linear combinations of the functions (71). Deleting the above functions a2, a2A, • • • from S, we have 5 = 7 + c/3 + eAfi, (3 = 61(63 + 0:2), where c and e are constants, and I is an invariant. Set /l = BX2^ + 5i.T2"-^T3 + \- Bn-lXoXz''-^ + BnXz^, and call a the coefficient of (94) XiXiXz ■ X2^''-^''-W~^ = (xi^xsyxi^'-^xi in xiXoXzcj). The coefficientf of (94) in R' of § 27 is Bi + a. Hence mhy a linear function of iMK'^-\ K^U, AK'^U, GK'^-K IFq^\ UQ,'^, ADQi'^ {i = 1, A, J; 1= 1, A, A, J). For n> 2, we may delete A from the part / of »S by use of EQi'^Qt"-'^, where E is given by (75). Without disturbing S we may delete 02(63 + 1) and its product by A from Si by use of j^2^+ij^n-z^ since the term of ^i'^f^ with the coefficient Si is the term of highest degree in x^ in ^a^^+H-Si-rs""^ + •••)• Since 5 + *Si is a linear combination of the functions (74), Si = S -\r tx{hz -\- aa) + haiai + Uhzaiao + ^463.4 + t^an (95) + ^A(63 + 1) + ^0(63 + 02)/. Apply (aia2a3)(6i62&3) to Bi, of the form (57). Hence (96) Si = ppai + ^0262 + pa2P + ra-z + sp, p = 61 + 03. Now aib2 occurs in S only in the terms J, AJ of I and in the part of (95) after S only in the last term, given by (72). In these the factors of aib2 are linearly independent. Hence ^0 = 0, I = x(A + 1). The coefficient of cti in ^i must vanish for 61 = a3, and Si itself if also a2 = 0. Hence c = t2 = X, h = ts = U = t, U = X + t, Si = x{A + 1 + 6163 + hia2 + aia2 + Aa;2) + ^.461(63 + 1) + t{hz +02 + hzaia2 + ^3^4 + 63A + a^A). Call € the coefficient in xiX2Xz4> of a-ia;2a:3 • .T2"^a-3^'^+"-3 = {X2WYX1X2XZ''--. In R' of § 27, the coefficient of this product is e + 5„_i. Hence Bn-i is of the form (57). Interchanging the subscripts 1 and 2 in Bn-i, we get »Si. Thus the coefficient of a^ in jSi vanishes for 63 = oi. Hence S = Si = 0. Any covariant with n > 2 differs 9G THE MADISON COLLOQUIUM. from one of rank > m by a linear combination of (i = 1, A, J; i = 1, A; / = 1, A, A, J). CovAEiANTS OF Odd Rank ??Z = 2/X + 1 > 1, §§ 30-31 30. Replacing xz by Xz + xi in the co variant (76), we get In XiX2Xz(t>, let g be the coefficient of {xiX^){x'^Xz)^~'^X2'^. The coefficient of the corresponding term of R' is g' ^ g -\- B, where B is that of .^2" in /i. Hence B is of the form (57). First, let n be odd. Then Si = Si-{- S under (50), so that S is a linear combination of functions (74) with a2(&3 +1) and its product by A deleted (§ 23). Thus S is the sum of the terms (95) after the first. Applying (aia2a3) (616263) to B, of the form (57), we see that *S is of the form (96). By these two results, 'S = ^(63 + 02 + 63aice2 + 63.4 4- 63A + a2A). If I is the coefficient of (x2X3-)^xz'^~^Xi in XiX2Xz(f>, that in R' is I' = I -\- nBn. Hence, for n odd, Bn is of the form (57). Inter- changing the subscripts 1, 2 in Bn, we get S. Thus the coefficient of ttz in »S vanishes for 63 = ai, so that i = 0. Any covariant with m and n odd differs from one of rank > m by a linear function of K^^L"" and Ai^"»X". 31. Finally, let m, be odd and n even. According as n = 4i' or 4v + 2, K'^Q.J or K'^-^MQi" is of the form (76) with 02 as the coefficient of ^2^xz'^. Hence we may delete the terms aoli in (85) and hence the terms aili in 5 of § 23. But (§ 30), B is of the form (57). Now 0361 occurs in J and A J of I and in 62 J of 62/2, having in these linearly independent multipliers. Hence 1= x{A+l) + 7/A, L2 = e-\-fA + gA. Since the coefficient of az in B shall vanish for 63 = ao, and B itself if also Ui = 0, we get ki = x = y = kz, ko = f = = e. im^ARIANTS AND NUMBER THEORY. 97 Thus S = x(A + 1 + A + 01^2 + biba + bia2) + hiaihihz + SrU + 1 + A + «ia2)&3 + hAh,(h, + 1). First, let n = 'iv -\- 2 and write 2;u + 1 for m. Then have d = (Ssifii + 1) and 026? as the coejSicients of ^2*^X3''. As in § 25, the coefficients of x, k^, g, h in (97) equal respectively (? + aoCA + &3 + 1), ttid-i- a^hz, Ad + a^d + 02/, Ad. The terms not containing d are combinations of the above a2li and 02(^3 + 1) of § 23. Any covariant with m = 2ju + 1 > 1, n = ^v -\- 2, differs from one of rank > m by a linear function of {i = 1, A; h = 1, A, J; 7 = 1, A, A). Next, let 71 = 4iV > 0. In the last two co variants of the theorem below, the coefficients of ^•i^'^'^'^Xz^" are a^bzibi + 1) and 5 = bz^zi^i + 1). We had reached covariants in which the corresponding coefficients are a^I and a^ibz +1)7. Thus we obtain the coefficient of ^4 in (97) and 5 + Aa2&3 + ciibibz, which equals the coefficient of g. We may therefore set k^ = g = 0. Subtracting covariants of the fourth and fifth types in the theorem, we may take as Si the function in § 24, without dis- turbing S. Applying (aia2)(6i62) to S and ^Si, we get 5„ and Bn-i- If I is the coefficient of XiX2^^^xz'^"*^"'~^ in XiX^Xzff), its coefficient in R' of § 30 is Z' = Z + 5„ + 5„_i. Thus Bn + Bn-i is of the form (57). By the coefficient of 0361, if4=0. Since the coefficient of az is zero for 63 = a^, we get x = k^ = tz = 0. Thus S= 0. Any cotariant ivith m = 2)u + 1 > 1, ?i = 41/ > 0, differs from one of rank > m by a linear function of K'"L'', AX«i", /Z'^Qo", ^X"-3Ql'^+^ iL^-^K~'-+\ G^Kqi'-^qi'^-^ FGq^'-^qi'^ a = 1, .4, A; Z = 1, A, J). 8 98 THE MADISON COLLOQUIUM. 32. We have now completed the proof of the theorem: As a fundamental system of modular covariants of the ternary quadratic form F ivith integral coefficients modulo 2, we may take F, its invariants A, A, J, its linear covariant L, its "polar" cubic covariant K, and the universal covariants Q\, Q2, L3. Incidentally, we have obtained a complete set of linearly independent covariants of each order and rank. We might then find a complete set of independent syzygies. Syzygies whose members are covariants of low rank are given in (78), (88), (91). 33. References on Modular Geometry. — Other aspects of the modular geometry of quadratic forms modulo 2 and, in particular, applications to theta functions have been considered by Coble. * For a treatment of non-homogeneous quadratic forms in x, y modulo p {p an odd prime), analogous to that of conies in elementary analytic geometry, but emplojdng only real points on the modular locus, see G. Arnoux, Essai de Geometric analytique modulaire, Paris, 1911. The earlier paper by Veblen and Bussey was cited in § 7. The paper by Mitchell was cited in § 3. Appli- cations of modular geometries have been made by Conwell.f The problem of coloring a map has been treated from the standpoint of modular geometry by Veblen. J * Transactions of the American Mathematical Society, vol. 14 (1913), pp. 241-276. ^Annals of Mathematics, ser. 2, vol. 11 (1910), pp. 60-76. t Annals of Mathematics, ser. 2, vol. 14 (1912), pp. 86-94. LECTURE V A THEORY OF PLANE CUBIC CURVES WITH A REAL INFLEXION POINT VALID IN ORDINARY AND IN MODULAR GEOMETRY 1. Normal Forvi of Cubic. — Consider a ternary cubic form C{x, y, z) with coefficients in a field F not having modulus 2 or 3. After applying a linear transformation with coefficients in F and of determinant unity, we may assume that (1, 0, 0) is an inflexion point. In particular, C lacks the term x^. If it lacks also x^y and x-z, its first partial derivatives vanish for y = z= 0. But we shall assume that the discriminant of C is not zero. Hence the coefficient of x- may be taken as the new variable y. At the inflexion point (1, 0, 0) the tangent ?/ = is to be an inflexion tangent, i. e., meet the cubic in a single point. Hence C lacks the term xz-. Thus C = x'-y + 2x(ay'^ + fiyz) + 0(^, z). Replacing x by x — ay — I3z, we see that x'^y is now the only term involving x. If y were a factor, the discriminant would be zero. Hence the term z^ occurs. Adding a suitable multiple of y to z, we get (1) C = x'y+gy' + kyh+dz' (5 + 0). 2. The Invariants s and t. — The Hessian of (1) is H = - Zbxh - hhf + 95(72/-3 + Wiyz^. The sides of an inflexion triangle form a degenerate cubic be- longing to the pencil of cubics kC + H- The latter has the factor z only when k = h = and the factor y — Iz only when' kl = 35 (as shown by the terms in a*-), where k is a root of Ar^ + 185/?A;2 + IQ^b-gk - 275-/r = 0. Before considering the factors involving a-, we note that the 99 100 THE SLADISOX COLLOQUIUM. coeflBcients of this quartic equation are the values which relative invariants of a general cubic assume for the case of our cubic (1). Indeed, a linear transformation of determinant unity which replaces C by a cubic C must replace H by the Hessian H' of C", and hence replace the inflexion triangle of C given by a root k of the quartic by that inflexion triangle of C which is given by the same number k. We denote the invariants by* (2) s= - 38h, t = - 1085-^. The above quartic now becomes (3) k'' - 6sk- - tk - 3s- = 0. The discriminant A of C is such that (4) 27A = f- QisK There are four distinct roots of (3) since its discriminant is - 273A2. Our earlier results are that kC + H has the factor z only when k = s = and the factor y — 3dk~h if ^ is a root =f= of (3) . It has the factor x — ry — pz if and only if 3p2 = k, 9d'-kr~ = s'- + tk/U, kp' - UpT = s, Qdkpr - 95 V2 - sk - ^/4 = 0. These conditions are satisfied if and only if k is a root of (3) and p = ^. = 0, 365-r- = -t (A: = 0), 3p2 = A-, 65A-r = p(il'2 - 3^) (k + 0). 3. The Four Inflexion Triangles. — First, let s = 0. Then i =}= by (4). The root k = gives the inflexion triangle with the sides (5) z= 0, x= ± ny mSW = - 0- * We have s = - 3*5', t = - 3«r, where S and T, given in Sahnon's Higher Plane Curves, p. 189, are the invariants of the general cubic with multinomial coeflBcients. INVARIANTS AND NUMBER THEORY. 101 Each root of k^ = t gives an inflexion triangle (6) y = J'^' ''=^''4^ + rV (3-366V = 0. Next, let 5 =# 0. Each root of (3) gives an inflexion triangle 35 ^ [kf , A;2 - Ss \ (7) y = -j^z, .^.= ±^-^^, + -^^yj. 4. The Parameter 8. — If we multiply x, y, z by p, p~^, p, we obtain from (1) a cubic with 5 replaced by dp^. If F is the field of all complex numbers, the field of all real numbers, or the finite field of the residues of integers modulo 3j + 2, a prime, every element is the cube of an element of the field [in the third case, e = {e~^y], so that the parameter d may be taken to be unity. Although we do not use the fact below, it is in place to state here that for all further fields a new invariant is needed to distinguish the classes of cubics (1). Indeed, two cubics (1), with coef- ficients in F and with the same invariants s and t and discrim- inants not zero, are equivalent under a linear transformation with coefiicients in F and having determinant unity if and only if the ratio of their 5's is the cube of an element of F. Criteria for 9, 3 or 1 Real Inflexion Points, §§ 5-9 5. Inflexion Points when s = 0. — Let /c be a fixed root of k^ = t. Let Ti and T2 be fixed roots of the equations at the end of (5) and (6). Then (Ti/r2)2 = - 3 = (1 + 2co)^ co= + CO + 1 = 0. Choose w so that Ti/r2 = 1 + 2a;. Denote the lines 2=0, X = ny, X = — Tiy in (5) by Xi, L2, L3. For each value of i = 0, 1, 2, denote the three lines (6) with k = Koi' by Ln, Liu Lzi, that with the lower sign being Lzi. Then the 9 inflexion points and the subscripts of the 4 inflexion lines through each are given in the following table: 102 THE MADISON COLLOQUIUM. (1,0,0) i (72, 1,0) (-TO, 1,0) (8) 10 11 12 1 20 21 22 30 31 32 (^^'^'35) 2 ( "^'^'35) 3 l2 l2 2,2-1 2,2-2 3,2-2 3, 2 - 1 In the last two columns, i has the values 0, 1, 2; while 2 — 1 or 2 — 2 is to be replaced bj^ the number 0, 1, 2 to which it is con- gruent modulo 3. WTien F is the field of all real numbers, k may be taken to be real, while just one of the numbers ti and T2 is real. Hence 3 and only 3 of the 9 inflexion points are real. The same residt is true if i^ is the field of the p residues of integers modulo p, where p is a prime 3 j + 2 > 2. For, k may be taken to be integral (§4), while co is imaginary and hence — 3 is a quadratic non-residue of j^- If — Hs a quadratic residue, ti is real and T2 imaginary. If — i is a non-residue, the reverse is true. Next, let 2> = 3j + 1, so that o) is real and hence — 3 a quad- ratic residue. By (5) and (6), ri and T2 are both real or both imaginary according as — Hs a quadratic residue or non-residue of p. Hence all 9 inflexion points are real if and only if — Hs both a square and a cube and hence a 6th power modulo p. If — ^ is a square but not a cube, only the first 3 inflexion points are real. If — ^s a quadratic non-residue, (1, 0, 0) is the only real inflexion point. A cubic with integral coefficients taken modulo p, a prime > 3, with at least one real inflexion point and loith invariant s = and invariant i =f= 0, has 9 real inflexion points if p = 3j -\- I and — t is a sixth power modulo p, a single real inflexion point if p = 3j -f 1 and — t is a quadratic non-residue of p, and exactly 3 real inflexion points in all of the remaining cases. For example, if p = 7 and 5 = 0, ^ 4= 0, there are 9 real in- flexion points only when i = — 1. Taking 5 = 3, xi = — 2, INVARIANTS AND NUIMBER THEORY. 103 r2 = + 1, K^ - 1, we get u=2. Thus x-y - 7/ + 32^ = has the 9 inflexion points (1, 0, 0), (1, 1, 0), (— 1, 1, 0), (-2, 1, 3 . 20, (2, 1, 3 • 20 a = 0, 1, 2). 6. Inflexion Points ichen 5 =t= 0, A =t= 0. — These are (1, 0, 0) and (9) / g-F 35 \ where A: ranges over the roots of the quartic (3) . We seek the number of real roots k for which V--^ is real. In order that the left member of (3) shall have the factors (10) A-- + wk + /, F - ick + m, it is necessary and sufficient that (11) I -\- m — w^ = — Qs, {I — m)w = t, Im = — 35-. Let ^ =1= (for / = see § 9). Then w + and (12) 21 = 2f^ - 6* + tjiv, 2m = vf- - %s - t/iv. Inserting these values into (II3), we get (13) w^ - 12sic^ + 48s-ir - f = 0. Set w~ = y -{- 4:S. Then (14) y^ = f- 645^ = 27A. For the rest of this section, let the field be that of the residues of integers modulo p, where y is an odd prime 3J + 2. Since any integer e has a unique cube root e~^ modulo p, there is a single real root 2/ of (14). First, let y -\- ^s be a quadratic residue of y. Then lo is real and hence also / and m. The product of the discriminants of the quadratic functions (10) is seen by (lli) and (II3) to equal (15) (u'2 - 4/)(^/>2 _ 4^„) _ _ 3(,j^,2 _ 4^)2 = _ 3^2 and hence is a quadratic non-residue of p. Thus a single one of the quadratics (10), say the first, has a discriminant which is a 104 THE MADISON COLLOQUirM. quadratic residue and hence has real roots. By (12i), 4l(iv^ - 4Z)w2 = _ 2iv^ - QicH + 365Z(;4 - if + 48stw - lUshv\ Adding the vanishing quantity (13), we see that (16) 4/(^2 - 4/)?(;2 = _ 3(m;3 _ g^^^, _^ ^)2^ Since w^ — 4^ is a quadratic residue and — 3 is a non-residue of p, it follows that Z is a non-residue. Hence a single one of the roots of the first quadratic (10), and hence a single one of the roots of the quartic (3), is the negative of a quadratic residue. Thus just two of the inflexion points (9) are real. Next, let 2/ + 4s be a quadratic non-residue of p. Then there is no factorization of the quartic (3) into real quadratic factors. Nor is there a real linear factor k — r and a real irreducible cubic factor. For, if so, the roots of the latter are of the form X, X^, X^" (cf. the first foot-note p. 37). Then (r-X)(r-XP)(r-XP'), P=(X-X^)(X^-X''')(X^'-X) = P^ (modp) are real, so that the discriminant of (3) is a quadratic residue. But this discriminant was seen to be — 3(81A)-, and — 3 is a non-residue. Hence (3) is irreducible modulo p. Thus (1, 0, 0) is the only real inflexion point. For p = Sj -\- 2 > 2, a cubic (1) icith stA 4= 0, has exactly three real inflexion points or a single one according as the real number 3A' + 4* is a quadratic residue or non-residue of p. 7. Cubic with stA 4= 0, p = 3j + 1. — Now — 3 is a quadratic residue of p and there are three real cube roots 1, co, co^ of unity modulo p. In this section we shall assume that A is a cube modulo p. Then there are three real roots yi of (14). At least one of the yi + 4s is a quadratic residue of p since n (yi + 4s) = 2/1^ + 64s3 = t'. If 2/1 + 4s is a quadratic residue, while 2/2 + 4s and 2/3 + 4s INVARIANTS AND NUMBER THEORY. 105 are non-residues, there is a single factorization of quartic (3) into real quadratics (10) and hence certainly not four real roots. The product (15) of the discriminants of the real quadratic factors is now a quadratic residue of p. If each were a residue, there would be four real roots. Hence each is a non-residue and there is no real root. There is a single real inflexion point if 2? = 3j + 1, 5/A 4= 0, A is a cube, and if the three values of 3A^ + 4* are not all quadratic residues of 2^- Next, let each yi -f- 4s be a quadratic residue of p. Then there are three ways of factoring quartic (3) into real quadratics (10). But a root common to two distinct real quadratics is real. Hence aU four roots are real. The discriminant of each quadratic (10) is therefore a quadratic residue of p. Hence, by (16), I is a quadratic residue of p; similarly for the constant term of each quadratic factor. Thus the negatives of the four roots are all quadratic residues or all non-residues. To decide between these alternatives, we need the actual roots. In Wi^ = yi + 45, let the signs of the Wj be chosen so that P - Wik + m; = (^ = 1, 2, 3) have a common root. As in (12), 2m i = IV p — Qs — tjiVi. For e 4= 1, we find by subtraction and cancellation of tvi — We that 2k = Wi -{- 10 e + t/{WilVc). Comparing the results for e = 2 and e = 3, we get (17) W1IV2WZ = t. Hence* the roots of (3) are |(U'i + Wi + U's), l{lOi — W2 — W3), i{— Wi-\- W2 — ICz), i(— Wi — ICo + W3). The product of the first and {i + l)th roots is seen to equal rrii In particular, we have deduced Euler's solution by the method of Descartes. 106 THE MADISON COLLOQUIUM. and hence is a quadratic residue. For given values of p, s, t, we can readily find by a table of indices the real values of the Wi and thus a real root and hence decide whether or not it (and hence each of the four roots) is the negative of a quadratic residue. However, changing our standpoint, we shall make an explicit determination of all sets s, t for which the quartic (3) has four real roots each the negative of a quadratic residue of y. By the definition of the u\^, or direct from (13), (19) 2tt'i2 = 125, ^loi^wi^ = 48^-, wxho^hoz^ = t\ Let w be a fixed integral root of co^ + co + 1 — (mod p). Then = (12^)- - 3(485^) = Swi^ - 2wiW = {ic-^ + coit?2" + orwz) (wr + (x>ho% + wwz) . Interchanging ^V2 and Wz, if necessary, we have (20) Wi^ + wit'o^ + u)hvz^ = (mod p). Conversely, if the ivi^ are any quadratic residues satisfying (20) and if we define s and t by (19i) and (17), we obtain a quartic (3) with the four real roots (18). If we permute wi, w^, lOz cyclically we obtain solutions of (20) leading to the same s and t and to the same four roots (18). Our first problem is therefore to find all sets of solutions of (20). To this end it is necessary to treat separately the cases — 1 a quadratic residue and — 1 a non-residue; viz., p = 12^+ 1 and p = 12(7+ 7 (since already p = 3J+ 1). First, let p = 12q -\- 1. Then — 1 = i^ (mod p), where i is an integer. Set 2p = «'i — iojicz, 2(r = Wi + iuwz. Then (20) becomes 4pa = — o)iV2^ = {ioi^Wify so that per must be a quadratic residue. Hence we may take INVAKIANTS AND NUMBER THEORY. 107 a = pV-, where p and / are integers not divisible by p. Then (21) Wi = p(l + V-), U'.2 = 2ioipl, Wz = 2co2p(l - /2). We must exclude the values of I which lead to equal values of two of the w'i", and hence to equal t//s, since the roots of (14) are incongruent. Now if any two of the w,^ in (20) are congruent, all three are congruent. But W\ = w^^ implies l-\-l^= ±2M, (/=F^co)^=w^ /=±2a; + cco2 (e^ ^ 1). The values Z- = 0, ± 1 make one of the lOi ^ 0. Hence we must exclude the 9 incongruent integral values of /: (22) 1 = 0, rt 1, d= 2', CO- ± fco, - co2 ± zco. Using the values (21), we get (23) 125 = p2{(l - co)(l + I') - Qd'P}, t = 2pH{l' - 1), (' 7 \ 2 ^ + ]lT^) • To make the negative of the last a square, we must take (25) p = - 2(1 + 2co2)r2 (r ^ 0). Now s, given bj- (23), is zero only when (26) Z = CO ± ior, — co =t ?co-. The desired sets s, t are given by (23) and (25), where r is any integer not divisible by p, tvhile I is any one of the 2^—13 positive integers < p not congruent modulo p to one of the 13 incongruent integers (22), (26). The minimum p is 37. Second, let p = 12g + 7. Then X- = — 1 (mod p) is irre- ducible. Its roots i and — i = i^ are Galois imaginaries. Set (27) TT = 2? + 1, a=p-l. There exists a linear function R of i with integral coefficients such that R""^ = 1, while no lower power of R is unity. Any function of i is zero or a power of i? and any integer is a power of 108 THE MADISON COLLOQUIUM. R", a primitive root of p. Hence we may set where ^ rj < a, ^ e < ira. Then (20) is equivalent to The last condition is equivalent to (28) e=2r] + y = -1, d" - io: = Rf-'l^ (/ odd), 2w22wi = 2R''^ + i?*+^"^2 - Rv^-f'i^ = R'''^J-U+i) I2\ (Rri-i+pU+i) 12 _|_ ]ipv-pj+(.f+i)/2y^ The last binomial is its own pih power and hence Is real. We desire that the root |2wi shall be the negative of a quadratic residue and hence a non-residue. Since R" is a primitive root of p, the condition is that j — (/ + l)/2 shall be odd: (30) / = 2/ - 1, j- 1 = odd. We must exclude the values making wi^ = ivi^: = 2R''l\ivi q= W2) = R^^+'^+J'^ =f 2co7^"''+<^/2 _ i^2pw-^ the second term having been simplified by use of Jln,r/2 = _ 1, RP- = R—, Completing the square of the first two terms, we get (Rn+'^U+^)/2 ^ ^^JlPv-'Tjr-y. = (^2^ l)R'^P^-y. Now co^ + 1 = — CO = (cioo^y, where c = 1 or — 1. Hence INVARIANTS AND NUMBER THEORY. 109 But (co + ico2)(co - iV) = - 1, CO + /co2 = i?'"^ (31) CO - ICO^ = - i?-'"^/2 (^ O^J(J)^ Hence we must exclude the four cases in which (32) rj^j+^{±v+l), i + i(±t'+7r+l) (mod tt), these four values being incongruent. No one of the w's in (29) is zero, since e is odd by (28), so that e # 0, 7r/2 (mod tt). By (190 and (17), 48* = (1 - o:){R-' + ii^pe) 4. Gco^Es.", (33) 4i = - iR'^^R^' - R^P'). Finally, we must here exclude the cases in which 5=0. Combining Swr = with (20), we obtain the necessary and sufficient condition Wi^ = C0W3- for 5 = 0. But Wi = d= co^ws, in connection with (29), gives R'{1 ± zco) = Rp'(- 1 ± zco), R'((^ ± zco2)2 = i^p^ Thus, by (31), the condition is that e =b lu = pe (mod tto-) or e = ± V (mod tt). Then, by (28), 7? is congruent modulo x to one of the values (32) decreased by ■7r/4. Hence the desired sets s, t are given by (33), subject to (28), in ivhich the 8 incongruent r]'s given by (32) and those values decreased by it 14: are excluded. In particular, p > 7. For p = 19, the only admissible pairs are , 5 = 2 '2'\ t = 6(- 2)3^ {1= 0,1, -..,8). For any I, the negatives of the roots of quartic (3) are the products of - 3 = 42, 4, 7 = 8^ - 8 = 72 by (- 2)^ and hence are quad- ratic residues of 19 since — 2=6^. For p = 31, the only pairs are 5=32^, i=5(_3)3i. s=-3'-\ f=13(-3)3' (; = 0, ••-, 15), the negatives of the roots of (3) being the products of 7, — 11, — 12, — 15 and — 3, 5, 9, — 11, respectively, by (— 3)^ and hence are quadratic residues of 31. 110 THE MADISON COLLOQUIUM. 8. Case 2^ = 3j + 1, stA =1= 0, A not a Cube.— The roots of (14) are now Galois imaginaries y, y^, y^'. As at the beginning of § 7, f-= (y-{- 4s) {y^ + 4s) (y^' + 4^) ^ (y + 4s)'+p+^\ Raise each member to the power (p — l)/2. We see that y + 4.9 is the square of an element, say w, of the Galois field of order p^. The first root (18) is ^{iv + w^ + w^') and equals its own pth power, and hence is real. This is not true of the remaining roots (18), since w'" =1= w, or since a real quadratic factor would imply that w is real. Hence the quartic has a single real root. For p = 7, the only cases in which the negative of the single real root is a quadratic residue are t = — 1 or 3, 5 = — 1, — 2, 3; t = 2, s arbitrary 4= 0. For p = 13, the only cases are ±t=4,5,Q; s = - 1,-3,4 (s^=-l); ±t=l,5,Q; 5=-2, -5, -6 (*'=5); and ±t=3,—s equals one of the preceding six values of s. 9. Cubic loith t = Q, s ^ 0. — In this case, (3) becomes (F - 3sY = 12s\ If there be a real root h, 3 is a quadratic residue of p, and /^2 =is^ / = 3 d= 2 Vs. First, let p = 3j -\- 2, so that — 3 is a quadratic non-residue of ;;. Then — 1 must be a non-residue of p and hence p = 12r + 11 . The product of the two Ts is — 3, so that a single value of F is a quadratic residue. Since the two real k's are of opposite sign, there is a single real root h whose negative is a quadratic residue. For / = 0, 5 # 0, and j) = 12r + 5, there is a single real inflexion point; for p = 12r +11, there are just three real inflexion points. Finally, let p = 3j + 1, so that — 3 is a quadratic residue of p. li p = 12r + 7, then 3 is a non-residue, and there is no real k and hence a single real inflexion point, li p = 12r + 1, the four roots h are all real or all imaginary. For p = 13, P = — 2* or — 55, and — /c is a quadratic residue if and only if k^ = 1, «3 = 8, 5 = 2, 5, 6. For p = 37, k^ = - 4* or 10^, and - k is a residue if and only if *** = 1. TOPICS IN THE THEORY OF FUNCTIONS OF SEVERAL COMPLEX VARIABLES BY WILLIAM FOGG OSGOOD CONTENTS LECTURE I A General Survey of the Field § 1. Analytic Functions of Several Complex Variables. . . Ill § 2. Jacobi's Theorem of Inversion and the Abelian Functions 113 § 3. Periodic Functions 114 § 4. The Theta Theorem 118 § 5. Automorphic Functions of Several Variables 122 § 6. Continuation. Hyperfuchsian and Hyperabelian Functions 125 § 7. Algebraic Functions of Two Variables 127 § 8. Analysis situs 132 LECTURE II Some General Theorems § L Definitions and Elementary Theorems 133 § 2. Line and Surface Integrals, Residues, and their Generalizations 135 § 3. The Space of Analysis, and Other Spaces 137 § 4. Rational and Algebraic Functions 142 § 5. Sufficient Conditions that a Function of Several Complex Variables be Analytic 142 § 6. Sufficient Conditions that a Function be Rational or Algebraic 143 § 7. On the Associated Radii of Convergence of a Power Series 145 § 8. Hartogs's Function Rx 150 § 9. On the Analvtic Continuation of a Logarithmic Potential 153 9 i 11 CONTENTS. § 10. The Representation of Certain Meromorphic Functions as Quotients 154 § 11. Integral Functions as Products of Prime Factors. . . . 156 LECTURE III Singular Points and Analytic Continuation § 1 . Introduction 160 § 2. Non-Essential Singularities 161 § 3. Essential Singularities 163 § 4. Removable Singularities 163 § 5. Analytic Continuation by Means of Cauchy's Integral Formula 165 § 6. Application to the Distribution of Singularities 169 § 7. Generalizations of the Theorem of § 5 171 § 8. Levi's Memoir of 1910 172 § 9. Continuation. Lacunary Spaces 175 § 10. Concerning the Boundary of the Domain of Definition of/(.T,2/) 177 §11. A Theorem Relating to Characteristic Surfaces 178 LECTURE IV Implicit Functions § 1. Weierstrass's Theorem of Factorization 181 § 2. A Tentative Generalization of the Theorem of Factorization 184 § 3. Algebroid Configurations 185 § 4. Continuation. The Branch Points of the Discrimi- nant 188 § 5. Single- Valued Functions on an Algebroid Configura- tion 190 § 6. Solution of a S^^stem of Analytic Equations. Weier- strass's Theorem 192 § 7. Continuation. A General Theorem 193 § 8. The Inverse of an Analytic Transformation 197 CONTENTS. Ill LECTURE V The Prime Function on an Algebraic Configuration § 1. The Algebraic Functions of Deficiency 1 and the Doubly Periodic Functions. GeneraHzations .... 199 § 2. The Prime Function Defined as a Limit 202 § 3. The Existence Theorems 204 § 4. Dependence on the Parameter 207 § 5. The Functions in the Automorphic Fundamental Domain 208 § 6. An Auxiliary Function 212 § 7. The Prime Function ^(t, t) 215 § 8. The Determination of Q{t, r) by Functional Equations 216 § 9. The Abelian Integrals in Terms of the Prime Function 218 § 10. The Integral of the Second Kind on i^ 221 § 11. The Integrands of the Integrals of the First Kind. . . 223 § 12. The Algebraic Functions 225 § 13. Parametric Representation of a Homogeneous Alge- braic Configuration 226 § 14. Linear Differential Equations on an Algebraic Con- figuration, and the Factor (p'(t) 228 TOPICS IN THE THEOEY OF FUNCTIONS OF SEVERAL COMPLEX VARIABLES BY WILLIAM FOGG OSGOOD LECTURE I A GENERAL SURVEY OF THE FIELD § 1. Analytic Functions of Seveeal Complex Variables In the decades which lay between Cauchy's prime and the beginnings of the modern French school, the theory of functions of a single complex variable made rapid progress, the chief advances taking place on German soil. Simultaneously with these developments, important problems in the theory of analytic functions of several complex variables were attacked and the theorems connected with them divined with an insight worthv of the genius of a Riemann and a Weierstrass. The elementary functions of several real variables admit exten- sion into the complex domain and are seen to be developable there by Taylor's theorem, — a result to which the elementary theory of infinite series and an obvious extension of Cauchy's integral formula alike lead. It was natural, then, to define a function of several complex variables generally with Weierstrass as one which can be de- veloped by Taylor's theorem in the neighborhood of any ordinary point of its domain of definition; or, following Cauchy, as one which is analytic in each variable separately and continuous in all taken at once.* * Cauchy, Turin memoir, 1831, = Exercices d'analyse, 2 (1841), p. 55; Jor- dan, Cours d'analyse, 1, 2d ed., 1893, §206. The condition of continuity is introduced to simplify the proofs. It is a consequence of the former condition ; 111 112 THE MADISON COLLOQUIUM. The Factorial Function and Analytic Continuatio?i. One of the problems with which mathematicians had occupied them- selves without obtaining satisfactory results was that of extending the definition of the function n ! to a continuous range of values for the argument. This question Weierstrass* took up, exam- ining the work of his predecessors and showing that a satisfactory solution could be reached on the basis of the principle of analytic continuation, the functions considered being dependent on several variables. Thus these functions contributed at that early time to the recognition of the importance of the conception of the monogenic analytic configuration. Existence Theorems. Cauchy had established the first existence theorems for ordinary differential equations and implicit func- tions.! In his further study of these problems he developed the method of power series and series majorantes.X The extension to the case of partial difterential equations was direct, and the results thus obtained were of importance. For, while much of the theory of these equations appeared plausible from geometric considerations of a somewhat crude sort or from analogy with special examples yielding an explicit solution, a secure foundation had hitherto been lacking. Weierstrass' s Theorem of Factorization. If a mathematical theory is to gain its independence and take its place among the powers, it must recognize its own peculiar problems and obtain methods for dealing with them. One of the earliest distinctive theorems which became known in the theory of functions of several complex variables is the theorem of factorization, due to Weierstrass. § cf . below, Lecture II, § 5. Such citations will be made in the following pages as H, § 5. In order not to interrupt the course of the general account with which we are now engaged, the consideration of a number of detailed consequences which follow from the definition will be postponed to a later paragraph; cf. II, §§1,2. * Journ.fiir Math., 51 (1856), p. 1; Werkc, 1, p. 153. t Cf. Enzyklopiidie der math. Wiss., II B 1, p. 103, and ibid. II A 4a, p. 201. J Turin memoir, 1831; Excrcices d'analyse, 1 (1840), p. 327. § Cf. IV, § 1. The theorem dates from 1860. FUNCTIONS OF SEVERAL COMPLEX VARLiBLES. 113 By the aid of this theorem he proved the extension of Riemann's theorem relating to removable singularities,* at least for the case that the given function can be expressed, in that part of the neighborhood of the given point where it is defined, as the quotient of two functions each analytic at the point. It would be of interest to know whether Weierstrass ever considered the theorem in its general form. I recall no passage in his writings which contains such a reference. Is it possible that the restricted form just mentioned was sufficient for all the applications of this important theorem which he met? § 2. Jacobi's Theorem of Inversion and the Abelian Functions Toward the close of the eighteenth century the way was paved, through Legendre's researches in the theory of the elliptic integrals, for some of the most important advances which have been made in analysis since the invention of the calculus, — those which cluster about the elliptic functions and their general- izations, the Abelian and the automorphic functions. Jacobi, following a line of thought which Abel had initiated, was led to formulate the problem of inversion which bears his name.f The first solutions of this problem which appeared, restricted to the case p = 2, — those of Gopel (1847) and Rosenhain (1846-51), — were based on the theta functions of two arguments.^ Weierstrass§ and Riemann|| arrived independently at solutions in the general case of the Abelian integrals corresponding to an * Cf. Ill, § 4. t Jacobi, Considerationes generales de transcendentibus Abelianis, 1832; Ges. Werke, 2, p. o. For a statement of the general problem cf. Neumann, Abelsche Integrale, 2d ed., 1884, Chs. 14, 15; Appell et Goursat, Fonctions algebriques, Ch. 10. For an account of the history of this problem cf. Krazer's Festrede: Zur Geschichte des Umkehrproblems der Integrale, Karlsruhe, 1908. t Jacobi and Gopel independently extended the elliptic thetas to the thetas of several arguments; cf. Krazer, 1. c, pp. 17, 18. § Beitrag zur Theorie der Abelschen Integrale, Braunsberg, 1849, = Werke, 1, p. Ill; Journ.fiir Math., 47 (1854) p. 289 = Werke, 1, p. 133; ibid., 52 (1856), p. 285= Werke, 1, p. 297. Also Werke, 4. II Journ.fiir Math., 54 (1857), pp. 101/155 = Werke, 1 ed., p. 81; 2d ed., p. 88. 114 THE M.VDISON COLLOQUIUM. arbitrary algebraic configuration. In these investigations both mathematicians were led to the study of the theta functions of p arguments, — in fact, Weierstrass, to whom the generalized thetas were at that time unknown, thus came to discover the form of these functions.* The Abelian functions themselves are not single-valued. They are the roots of algebraic equations of degree p, whose coefficients are single-valued functions having only non-essential singularities in the finite region of the space of their p complex arguments and admitting 2p independent periods; cf. § 3. Here, then, is a general class of functions of several variables, to which Jacobi's problem of inversion has directly led, — the class which corresponds to the doubl}^ periodic functions of a single variable. § 3. Periodic Functions To state more precisely what is meant by periodicity, it is this. The function f(zi, • • • , Zn) is said to admit the periodf (P) = (Pi, •••,Pn)if /(Zl + Pi, 22 + P2, •••,Zn+ Pn) = /(Zl, ' ' *, Zn), where Pi, • • • , Pn are constants. AYe shall restrict ourselves here, unless the contrary is explicitly stated, to functions which are single-valued and have no other than non-essential singularities (III, § 2) in the finite region of space. If (P) and (Q) are two periods, then (P) + (Q) = (Pi + Qu •••, Pn-\-Qn) is evidently also a period. Moreover, (— P) = (—Pi, • • •, —Pn) is a period. A function /(zi, • • •, z„) is said to be /j-fold periodic if there exist k periods (P'), iP"), • • • (P^^'O, and no fewer, in terms of * For their definition cf. § 3. t Weierstrass uses the term system of periods (Periodensystem), i. e., simultatieous system of periods, to denote tliis complex, which may be thought of as a vector in space of 2p dimensions. The briefer term period would seem to suffice. FUNCTIONS OF SEVERAL COMPLEX VARL-VBLES. llo which every period (P) can be expressed linearly with integral coeflBcients : (P) = m'(P') + m"(P") + h in^'^P^''^). Such a set of periods is called a primitive scheme, or set, of periods.* A periodic function which is a constant or which depends on fewer than ?? arguments will evidently not come under this definition. This will also be the case if, on making a suitable non-singular linear transformation of the arguments, f(zi, • • • , 2„) goes over into a function of fewer than n arguments. All other periodic functions do come under this definition, the functions excluded being precisely those which admit infinitely small periods. It is a theorem due to Riemannf that a A'-fold periodic function of ^^-independent variables cannot exist! when k > 229. On the other hand, the Abelian functions have led to 2p-fold periodic functions of j^ complex arguments, and such functions can also be formed by means of quotients of theta functions of p arguments. Theta Functions idth Several Arguments. — The fundamental theta function of a single argument! can be defined by a series as follows: t?(w) = d{u, a) = C T, e°"'^2n«^ C + 0, where . a = r -\- si and r = ^(a) < * I avoid the term primitive system of periods because of the confusion which would thus be introduced, due to the other sense, above mentioned, in which the words system of 'periods are used. \Journ.Sur Math., 71 (1859), p. 197=Werke, 1 ed., p. 276; 2d ed., p. 294. Cf. also Weierstrass, Berliner Monatsber., 187G, p. 680 = Werke, 2, p. 55. t The maximum number of periods which an integral function can have is p. Hermite, in Lacroix's Calcul differentiel et calcul integral, vol. 2, 6th ed., 1862, p. 390. § This function appears in Fourier's Theorie analytique de la chaleur, 1822, p. 333. It is usually thought of as due to Jacobi, who was the first to recognize its importance in the theory of the elliptic functions; Fundamenta nova, 1829, =Werke, 1, p. 228. 116 THE MADISON COLLOQUIUM. It has the properties: ^{u + Tvi) = ^{u), i?(w+ a) = e-2"-«t?(w); and it has, moreover, a single root of the first order in the parallel- ogram F, two sides of which are the vectors (0, wi) and (0, a). By means of this function, doubly periodic functions can be formed as follows. Let cei, • • • , a„, /3i, • • -, fin be any 2n points so chosen that n n H(Xk = S fik, and that, furthermore, the points of the parallelogram F that are congruent to them are distinct. Then the quotient i}{u -\- ai) • • • t?(M + OCn) HU + iSl) • • • HU + fin) will evidently represent a doubly periodic function with the periods iri and a. The fundamental theta function of p arguments is given by the following series: p r+2 S n*MA ^(ui, '•■,up) = CZe *=* , C4=0, where p r = r(wi, • • -, np) = J2 akinkUi, aki = aikr k, /=1 (Ikl = Tkl + iSkl, and the real part of T{xi, • • •, Xp), where xi, - ■ •, Xp denote real variables, namely p Z) TklXkXi, k,l=\ is a definite negative quadratic form. The function has the following properties, readily deducible FUNCTIONS OF SEVERAL COMPLEX VAmABLES. 117 from the series,* which we write at length for the typical case p = 3. «?(?/! + iri, u.>, Uz) = §{11.1, 112, W3), t?(wi, «2 + tt/, Uz) = d(ui, 7/2, Uz), ■&{ui, ?/2, iiz + Tri) = ??(2/i, U2, Uz), ^{Ul + «ii, «2 + 0-21, Us + fi3l) = e~^"'~''"??(«l, ?/2, ?<3), t?(Wi + «12, ?<2 + «22, W3 + a32) = e~^"-~''*'?9(?/l, Wo, ws), t?(wi + ai3, U2 + ^23, W3 + a33) = e~^""'~''''^(ui, U-i, Uz). The vectors in 2p-dimensional space corresponding to the 2p columns in the array ri (1) 7r^ iri Oil an an Cf21 O22 Q23 «31 O32 «33 form the edges of a true prismatoid, F, and a periodic function corresponding to F can be formed as follows. Let 2pn = Gti complex numbers an, fiki, k = I, • • -, p = S, I = I, • • • , n, he so chosen that «11 + • ■ -\- OCln = jSll + • • • + ^m, CX21 + • • + (Xin = ^21 + • ■ ■ + ^2n, «31 + • • • + azn = /331 + • ■ ■ + ^Zn, but that these numbers are otherwise non-specialized. Then the quotient t^(Ui-\-aii, U2-\-a21, ^3 + 0:31) • ' • ^(Ul+ain, U2 + a2n, M3+Q:3n) «?(Wl+|8ll, U2+^2l, Uz+M • • • ^{Ul+^ln, W2+j82n, Uz+^Zn) will represent a function admitting as a primitive scheme of periods the above scheme (1). It is sufficient to take n = 2. As regards the proof of this theorem, it is clear that the above quotient admits each period of the scheme (1); but it is not * Cf. Ivrazer, Lehrbuch der Thetafunktionen, Chap. 1. lis THE MADISON COLLOQUIUM. clear that the a's and /3's can be so chosen that this scheme is primitive for the function. This is, however, the case.* A second mode of obtaining 2p-fold periodic functions belong- ing to the scheme (1) is as follows. The functions 52 log ^ dzk dzi k, I = 1, '--yih obviously admit the periods of (1), and it is readily shown that they admit only such periods as are expressible linearly with integral coefficients in terms of these.f And now it can be proven that a linear combination of the above functions can be so chosen as to yield a function belonging to the scheme (1). This statement is made by Wirtinger, 1. c, but the proof is far from obvious. t The number of essential constants on which an algebraic con- figuration of deficiency p > 1 depends is 32:* — 3, — the so-called moduli. For p — 2 and 2^ = 3 this number is the same as the number of complex constants in the theta function, namely ^p{p + 1). But for 2? > 3 the latter number is larger, and hence the Abelian functions of p arguments, — or rather the symmetric functions of their multiple determinations, — are not the most general 2p-fold periodic functions. § 4. The Theta Tiieokem. Can all 22J-fold periodic functions with only non-essential singu- larities in the finite region be expressed in terms of theta functions of p arguments? The answer to this question is affirmative, and is the noted theta theorem due to Riemann and Weierstrass. At first sight a mere count of constants appears to discredit the theorem. For the general theta function of 2> arguments depends on but ^pip + 1) complex, or x^ip + 1) real constants, namely, the Uki subject to the equations a^-^ = aik, while the region of 223-dimensional space which is the analogue of the *Cf. a forthcoming paper by the author. (Note of December 29, 1913.) t WirtinfTor, Monalshefte f. Math. u. Phys., G (1895), p. 96, § IG. t Cf. a forthcoming paper by the author. (Note of January IS, 1914.) FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 119 parallelogram of periods for ^J = 1 and which forms a funda- mental region for the function, — the prismatoid, /', — depends, after reduction to normal form, as we shall presently see, on p^ complex, or 2jr real constants. With reference to this normal form, let 21, • • • ,Zp be the original arguments and let the original 2p periods, which are linearly inde- pendent, be WTitten in the columns of the following array: (2) Then at least one of the p-rowed determinants taken from the matrix of the 2p2 ^^'g corresponding to the scheme (2) will be dif- ferent from 0.* Let this be the determinant ± Scon • • • copp. If now we set TviZk = OikXllX + • • • + CO^.p?/p, Zi cou • • • COip £0u • • • Wl'p 22 W21 • • • C02p W21 • • • W2p 2p Wpi • •• COpp COpl • • • • 03pp the scheme of periods for the transformed function /(zi, = F{u\, • ' ' , Up) will be as follows: Zp) Ul Wl, 0, ' ' ' ) an. •••) Clip H2 0, iri, ' ' ' ) 021, ' • , (l2p Up 0, 0, ... > 7^^ dpi, ' ' ' } 0,pp Thus we have in the aus p^ complex, or 2/?^ real constants. If these be given non-specialized values, we are led to a true 2/>dimensional prismatoid. To any n-dimensional prismatoid F correspond real analytic functions of n real variables with n periods, for which F is a fundamental domain. If, then, in the case before us, the most *This theorem is contained in a paper by Kronecker, Berliner Sitzimgsber., 29, (1884), p. 1071. Its proof follows, however, readily on developing system- atically the elements of the periodic functions of several complex variables from a geometric standpoint. 120 THE MADISON COLLOQUIUM. general 22>fold periodic analytic functions of y complex variables are to be represented by means of quotients of thetas with p arguments, this means that the prismatoid is here subject to essential restrictions, since ^(p + 1) < 2;;-. That this is, in fact, the case was discovered independently b}' Riemann and Weierstrass, and thus the first step was taken toward the establishment of the theta theorem. Riemann never published a proof of the theorem. He com- municated his results to Hermite* in 18G0. Weierstrass's proof was not given in detail till the appearance of his collected works, f though he had published a number of notes bearing on a proof, and had stated the theorem in a letter to Borchardt.| In the early eighties Poincare and Picard§ constructed proofs of the theorem, which, it turned out, were essentially the same as Weierstrass's. Appell|| gave a proof in 1891 along different lines. Then came a proof by Wirtinger,^ which has much in common with Weierstrass's proof, at that time unpublished. Shortly after, Poincare** gave a new proof, in which the method is that of potential functions in hyperspace. Kroneckerff had already surmized that this method would lead to fruitful results in the theory of functions of several complex variables. |t Poin- care had used this method in an earlier paper, in proving the theorem that a function of two complex variables which has no other than non-essential singularities in finite space, can be expressed as the quotient of two integral functions, and that this quotient, moreover, at any point at which both numerator and * Cf. Lacroix, Calcul diffcrentiel et calcul integral, vol. 2, 6th ed., 1862, J). 390. t Werke, 3, 1903, p. 53. tJourn.fiir Math., 89 (1880), p. 8 = Werke 2, p. 133. Berliner Monats- berichte, 1869, p. 855= Werke 1, p. 46. § C. R., 97 (1883), p. 1284. Poincar6, Acta, 22 (1898), p. 90. II C. R., 110 (1890), pp. 32, 181; Journ. de Math. (4), 7 (1891), p. 157. t Monatsheflef. Math. u. Phys., 6 (1895), p. 69. Cf. also ibid., 7 (1896), p. 1. ** Acta, 22 (1898), p. 89. Cf. also ibid., 26 (1902), p. 43. tt Berliner Monal.sberichte, 1869, pp. 159, 688 = Werke, 1, p. 198. Xt Cf. two mcmoins by Baker, Transactions Cambridge Phil. Soc, 18 (1900), p. 408, and Proceedings London Math. Soc. 2), 1 (1904), p. 14. FUNCTIONS OF SEVERAL COMPLEX VARL\BLES. 121 denominator vanish, will be in reduced form.* It is this theorem, too, on which Appell's proof cited above rests. All of these proofs involve a considerable amount of analytical developments. Weierstrass was led, in the course of his analysis, — and it may be remarked in passing that he edited his proof with minute care, — to emphasize the importance of an accurate definition of the monogenic analytic configuration of the mth grade (Stufe) in the domain of n complex variables. He points out that it will not do to start with the points for which certain of the coordinates chosen as dependent variables are analytic in the remaining coordinates considered as independent variables, and then adjoin all limiting points to the set thus obtained. For, in the case of two variables, he says, it may happen that one would thus obtain all the points of space, f Furthermore, in the proof as Weierstrass originally conceived it, — the final proof which appeared in his collected works is modified in essential respects, — two general theorems relating to periodic functions play an essential role. They are these, i I. Any 2p-fold periodic function of i) variables is an algebraic function of ;; independent 2^>fold periodic functions belonging to the same prismatoid. Or, otherwise expressed: Betw^een any p + 1 2p-fold periodic functions of p variables there exists an algebraic relation. II. Any 2p4o\d periodic function of p variables is expressible rationally in terms oi p -\- I suitably chosen 22>fold periodic functions belonging to the same prismatoid. These theorems have been generalized by Picard and Wirtinger for automorphic functions of several variables; cf. §§5, G. Poincare's potential functions undoubtedly form a powerful instrument of analysis in dealing with the singularities of func- *Cf. IV, §1. t Werke, 3, p. 96. X Berliner Monatsberichle, 1869, p. 855= Werke, 2, p. 46. These theorems have been treated by Poincare, C. R., 124 (1897), p. 1407; Wirtinger, Sitzungs- ber. der Wiener Akad., 108 (1899), p. 1239; and Bkimenthal, Math. Ann., 58 (1904), p. 497; cf. also Math. Ann., 56 (1903), pp. 510, 512. 122 THE MADISON COLLOQUIUM. tions of several complex variables. He carries his proof through only to the point of showing that the given function can be \VTitten as the quotient of two Jacobian functions. The latter functions are defined as follows. Jacobian Functions. Let co^^, a= 1, •••, j^l — 1> '••> 2p, be a primitive scheme of periods, and let f{zi, • • • , Zn) be an integral function of its p arguments. If, for every period from this scheme, a relation of the form holds: f(z, H- co,3, • • •, s,, + a;,^) = e^^^'VC^u '', ^p), where Lp{z) is a linear (homogeneous or non-homogeneous, but integral) function of Zi, • • • , Zp, then/ is called a Jacobian function. The Jacobian functions have been studied at length in two memoirs by Frobenius,* and in a paper by Wirtinger.f A Jacobian function can be expressed in terms of theta functions of p arguments. § 5. AuTOMORPHic Functions of Several Variables The brilliant results obtained b}^ Klein and Poincare in the early eighties, in their researches relating to the automorphic functions of a single complex variable turned the attention of mathematicians towards functions of several complex variables which admit a discrete group of linear transformations into themselves, and we find from that time to the present day a steady stream of papers in this field. Here, however, at the very threshold of the subject, two types of groups present themselves, corresponding on the one hand formally to the linear transformations of projective space: (U ( I 0''^ + b'y + C a"x + h"y + c" \ ■^ V'y\ax-\-hy-\-c' ax + hy + c J' and on the other, to those of the space of analysis: * Journ. fur Math., 97 (1884), p. 16 and p. 188. t Monalshcfle fiir Math. u. Phys., 7 (1896), p. 1. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 123 (2) Y I ax+ b ay -\- ^ \ V'y\ cx+d' 72/ + 5 /' / I a'y + b' a'x + ^' \ X'^l c'y+d" y'x + b')' Hypermodular Functions. — The first papers to appear in this field dealt with groups of the type (1). Picard* began by in- vestigating a class of functions of two independent variables analogous to the elliptic modular functions. It is a familiar fact that a hypergeometric integral r dt }/t(t -l){t-x)' where g, h denote any two of the four points 0, 1, oo, x, is a solution of the linear differential equation Let ui, C02 be two linearly independent solutions of this equa- tion, and set 77= m- Then the equation f(x) = u defines x as a function of u, and this function is analytic through- out the whole upper half of the w-plane, but cannot be continued analytically beyond this region. Picard passes to analogous functions of x, y, namely those defined by one of the integrals r dt ,v Vt{t- m-x)(t-y) where g, h denote any two of the five points, 0, 1, oo , x, y. These functions satisfy a simultaneous system of linear partial differ- * C. R., 93 (1881), p. 835; ibid., 94 (1882), p. 579; Acta, 2 (1883), p. 114. Alezais, " Sur une classe de fonctions hyperfuchsiennes," etc., Paris, 1901. 10 124 THE MADISON COLLOQUIUM. ential equations of the second order, the coeflBcients being polynomials in x and y, at most of the third degree, with integral coefficients. These equations admit three linearly independent solutions, coi, C02, C03. If the latter be suitably chosen and their ratios set equal to two new variables, CtJo Oiz - w. — x Oil Wl then these equations deJSne x and y as single-valued functions of w, v. The domain of definition, D, is that part of the four- dimensional space of the variables u = u' -\- iu", v = v' -{■ iv", in which 2v' + u'^ + u"^ < 0. The proof is given by means of the solution of Jacobi's problem of inversion for p = 3; cf. § 2. Picard shows that the functions thus obtained admit a properly discontinuous group of linear transformations of the type (1) which carry D over into itself, the coefficients being of the form k -\- ZX, where k and I are integers, and X is a complex cube root of unity. These transformations are closely related to those of a ternary group: X= Mix+P,y-\-R^z, Y= M2X+P2y+R2Z, Z = Ahx + P,y + R^z, — the coefficients here being also rational functions of X, — which leave the Hermiteian form xx-{- yy -\- zz unchanged, where x denotes, as usual, the conjugate of x. Generalizations of Riemann's P-Function. The investigations on which we have just reported suggest, tlirough the hyper- geometric integral and the hypergeometric differential equation FUNCTIONS OF SEVEEAL COMPLEX VARIABLES. 125 mentioned at the outset, Riemann's researches on binary families. In fact, Appell* had just been engaged in extending these results to quaternary families of functions of two independent variables, and Picardf had himself been working in the same field. § 6. Continuation. Hyperfuchsian and Hyperabelian Functions A further paper of Picard| deals with functions F(u, v) mero- morphic in their domain of definition, D, which consists of the interior of the hypersphere u'^ + u"^ + v'- + v"- < 1, and admitting a group of transformations into themselves of type (1). The fundamental domain of the group lies wholly within D. There is an allied system of simultaneous linear partial differential equations of the second order. Between three such functions there always exists an algebraic relation, — a property corresponding to Weierstrass's first theorem concerning periodic functions (§ 4, end), and these functions serve to uniformize such an algebraic configuration. Double integrals on the corresponding algebraic configuration! are studied, being uniformized as functions of u, v, and in this investigation we have a forerunner of Picard's researches on algebraic functions of two variables, to which we shall presently turn. Functions of the classes hitherto treated, namely, those which admit a group of transformations of type (1), are called hyper- fuchsian functions. \\ The definition is not restricted to functions * C. R., 90 (1880), pp. 296, 731; Journ. de Math. (3), 8 (1882), p. 173. tC. R., 90 (1880), pp. 1118, 1267; Ann. Ec. Norm. (2), 10 (1881), p. 305. t C. R., 96 (1883), p. 320; C. R., 99 (1884), p. 852. We note here a paper by Poincare, C. R., 94 (1882), p. 840, in which automorphic functions of two variables are obtained from the theory of numbers. Cf. also papers by Picard, Acta, 1 (1883), p. 297; ibid., 5 (1884), p. 121; Ann. Ec. Norm. (3), 2 (1885), p. 357. § Cf. Ill, § 1. 11 Picard, Acta, 5 (1884), p. 121. 126 THE MADISON COLLOQUIUM. for which D is the hypersphere, but includes at least all functions admitting a properly discontinuous group of type (1) and mero- morphic in a domain D defined by a relation giti', u", /, /') < 0, where gr is a quadratic polynomial. Moreover, the functions cannot be continued analytically beyond D. In this same year Picard* began the investigation of functions which admit a group of transformations of type (2). These functions he denoted as hyperabelian functions, since the first problem which he was led to study concerning them was one related to the Abelian thetas and the Abelian modular functions, 2? = 2. The classes discussed yielded functions with properties analogous to those of the hyperfuchsian functions. Generalizations. In a systematic development of the theory of the automorphic functions of several complex variables a question of first importance is that of the existence of a funda- mental domain belonging to a properly discontinuous group. A solution of this problem for such groups of projective trans- formations in n variables, — groups of type (1), — has been given bv Hurwitz.t The extension of the two theorems of Weierstrass, § 4, for the case of automorphic functions in n variables has been treated by Wirtingerf by the aid of methods of the general theory of functions. A systematic generalization of the theory of a class of hyper- abelian functions was outlined by Hilbert and elaborated by * Notes in the Cotnptes Rendus for 1884, followed by a systematic presen- tation in Journ. de Math. (4), 1 (1885), p. 87. Cf. further Bourget, Toulouse Ann., 12 (1898), p. D 1; Humbert, Journ. de Math. (5), 5, 6, 7, 9, 10 (1899- 1904), and (6), 2 (1906). ^Malh. Ann., 61 (1905), p. 325. tSitzungsber. der Wiener Akad., 108 (1899), p. 1239. For the special case of hyperabelian functions of n variables cf. Blumenthal, Math. Ann., 56 (1903), p. 510; ibid., 58 (1904), p. 497. Picard had long since used the second theorem, stated for automorphic functions of two variables; cf. Journ. de Math. (4), 1 (1885), p. 313. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 127 Blumenthal.* The group is that in which aixi+ ^i JiXi -f- Oi the coefficients being taken as follows. An algebraic domain of rationality is assumed as given, R = k, where k denotes a root of an irreducible algebraic equation in the natural domain, R = 1. Furthermore, all the roots A-, A-', • • •, /t^"~^^ shall be real. The coefficients ai, • • •, 61 are taken in R = k, and the coefficients ai, ' ■ • ,bi are the corresponding numbers of the domain R=k''^~^\ Finally, aidi — /3i7i is a totally positive unit of the domain R = k. The subject of automorphic groups in one and more variables has been treated systematically by Fubini.t § 7. Algebraic Functions of Two Variables The impetus given to the study of the algebraic plane curves and the geometry on them, through the researches of Pliicker, Cayley, and Clebsch, in connection with the theory of the algebraic functions and the Abelian integrals as developed by Riemann, early made itself felt in the study of algebraic surfaces and algebraic functions of two variables. Thus we find a paper by Clebsch J of the year 1868, in which he discovers an invariant of an algebraic surface analogous to the deficiency p of an alge- braic curve. The latter invariant may be defined as the number of essential constants in the general integral of the first kind, i. e., in the everywhere finite integral, and this integral can be written in the form / *Cf. preceding reference. Furthermore Hecke, Gottinger Dissertation, 1910. t Introduzione alia teoria dei gruppi discontinui e delle funzioni automorfe, 1908. t C. R., 67 (1868), p. 1238. Clebsch had only the adjoint Q's of degree m — 4. The everywhere finite double integral is due to Noether, Math. Ann., 2 (1870), p. 293. 128 THE MADISON COLLOQUIUM. where f{x,y) = is the equation of the ground curve Cm, assumed irreducible, and Q{x, y) is an adjoint polynomial of degree m — 3. If, in particular. Cm has only ordinary double points, Q = is any Cm-z that passes through these points. Consider now an irreducible algebraic surface f{x, y,z) = of degree m vfith. only ordinary multiple lines and isolated multiple points. Then the double integral (II, § 2) // Q(x, y,z) -ax ay taken over an arbitrary regular surface, open or closed, lying in the four-dimensional Riemann manifold corresponding to the function z of x, y defined by the equation / = 0, will remain finite provided Q{x, y,z) = is an adjoint surface of degree m — 4, i. e., a surface which passes through the multiple lines and has a multiple line of order A; — 1 at least in every multiple line of /of order k; and which moreover has a multiple point of order q — 2 Sit least in every isolated multiple point of / of order q* Such an integral is called a double integral of the first kind. The number of linearly independent integrals of this class, i. e., the number of essential constants in the adjoint polynomial Q{x, y, z) is called the deficiency, or more precisely, the geometrical de- ficiency, — Flachengeschlecht,t genre geometrique, — in distinc- tion from the numerical deficiency presently to be considered, and is denoted by 'pg. It is an invariant under the group of birational transformations of the surface: X = ri(.T, y, z), (A) Y = r.ix, y, z), Z = Tsix, y, z),-^ x = Ri(X, Y,Z), y = R,(X, Y, Z), z= R,(X, Y,Z).. In case the surface / has no multiple lines or points, (m- l)(m- 2)(m- 3) Pa 6 * Cf. Pifard et Simart, Fonctions alg6briques de deux variables, vol. 1, 1897, ch. 7; in particular, p. 189. t The invariant is due to Clebsch; the name to Noether. FUNCTIOXS OF SEVERAL COMPLEX VARLiBLES. 129 The Second Deficiency. There is a second numerical invariant which can be defined as follows. Consider the linear family of adjoint surfaces of degree ??i — 4: Q{x, ?/, s) = aiQi + 0:2^2 +••• + oip^Qv,' These surfaces cut the ground surface / = in certain fixed curves, — including always the multiple curves of /, — and a variable curve, I. This latter curve will, in general, be irreducible, and we assume the non-specialized case. It is a twisted space curve, and it has, as such, a definite deficiency, which can be defined, for example, as the deficiency of the Riemann's surface corresponding to the curve. This deficiency is the same in general for the different curves of the family, and it is this number, 2?^^^ which is called the second or numerical deficiency, — Kur- vengeschlecht,* le second genre. It is an invariant under the group of birational transformations, (^).t The Line Integral. There is another generalization of the Abelian integrals possible for the algebraic functions of two variables, namely,1: SPdx+Qdy, where P and Q are rational functions of x, y, z, the third variable being a root of the irreducible algebraic equation f{x, y, z) = 0, and where, moreover, the condition of integrability is satisfied: dP _ dQ dy dx ' * This invariant is due to Noether, Math. Ann., 8 (1875), p. 520. t Cf. Picard et Simart, I. c, p. 206. Noether introduced a further invariant, p^^^, namely, the number of variable points of intersection of two curves I. In general, p<^^ = p^^> — 1, but for special surfaces p^^> < p'^' — 1. Cf. Picard et Simart, ibid., p. 209. J Picard, Journ. de Math. (4), 1 (1885), p. 281; ibid. (4), 5 (1899), p. 135. The latter paper is the memoir to which the prize of the Paris Academy of Sciences was awarded. It forms the foundation of the later presentation of the theory of Picard and Simart, Fonctions alg^briques de deux variables, Paris, 1897-1900. 130 THE MADISON COLLOQUIUM. Such an integral is a function of two independent variables, and these may be taken as x, y or y, z or z, a: A division of such integrals into three classes, corresponding to the three classes of Abelian integrals, at once suggests itself. In the first paper above referred to Picard studies the integrals of the first class, namely, the everywhere finite integrals.* He finds here a situation diametrically opposite to that in the case of the Abelian integrals. If f{x, y) = is an irreducible alge- braic equation of degree greater than 2, there will in general exist integrals of the first class corresponding to it; it is only when the curve is highly specialized (unicursal) that this is not the case. To the non-specialized algebraic surface of arbitrary degree, however, there correspond no integrals of the first kind with the trivial exception of a constant. A special class of surfaces and integrals is treated, the former being those which can be uni- formized by means of quadruply periodic functions of two inde- pendent variables. It was in these papers that Picard began the study of questions relating to the connectivity of the surfaces which present them- selves. The points of an algebraic surface fill a four-dimensional region, — be that region assumed as a four-dimensional manifold in space of six or more dimensions, or as a multiple-sheeted Riemann manifold, or as a fundamental domain, for which the parallelogram of periods is the prototype. In this four- dimensional manifold the linear cycles (closed curves) and the two-dimensional cycles (closed surfaces) are of especial im- portance. Picard finds the striking result that, in the case of a non-specialized algebraic surface, any linear cycle can be drawn together continuously to a point. This fact explains, — or is explained by, — the non-existence of integrals of the first class on such a surface. On the other hand, a non-specialized algebraic surface does * Picard's first publication relating to the integrals of the second class appeared in the Comples Rendus, 100 (1885), p. 843. FUNCTIONS OF SEVERAL COMPLEX VARL\BLES. 131 admit two-dimensional cycles, and it is these that form the analogue of the linear cycles in the case of the algebraic functions of a single variable. With these are connected the double integrals of Noether. The methods employed in these early geometric investigations are largely those of intuition and analogy, Picard recognizes this fact, but points out that his chief object was to throw light on a theory at that time wholly new. Geometry on Algebraic Curves and Surfaces. The purely al- gebraic theory of the geometry of systems of points on algebraic curves has been extended to algebraic surfaces and systems of curves lying on them.* The Point of View of the Theory of Numbers. The methods of the theory of algebraic numbers, first extended to the algebraic functions of a single variable, have been used by Henself for the study of algebraic functions of two variables. In his treat- ment of the theory of the algebraic functions of a single variable Weierstrass had used purely algebraic methods. Hensel de- scribes his own methods for algebraic functions of two variables as the direct generalization of Weierstrass's methods. In a preliminary study of these functions Hensel deduces series developments which apply to the neighborhood of a branch-line or of a multiple-line of the surface. The form of the development in the neighborhood of a finite point, which we will take as the origin (0, 0, 0), is the following: z = e^(xXy - yoY'" + e,(x){y - y,)"^ + • • •, where 6 is a positive integer. The coeflScients ek{x) and the variable yo are analytic functions of ^, where and a is a positive integer. In fact, the equation of the branch * Xoether, Math. Ann., 2 (1870), p. 293; ibid., 3 (1871), pp. IGl, 547; ibid., 8 (1875), p. 495. Picard et Simart, Fonctions algebriques de deux variables, vol. 2. t Acta, 23 (1900), p. 339; Jahresber. D. M.-V., 8 (1899), p. 221. 132 THE MADISOX COLLOQUIUM. of the discriminant under consideration is § 8. Analysis Situs In closing we refer briefly to the subject of analysis situs in the geometry of n dimensions. Riemann was the first to recognize the importance of this subject for the surfaces which bear his name. He had also thought about the problem for higher manifolds.* Bettif considered the simple closed cycles of one dimension (curves), of two dimensions (surfaces), and, generally, of ??i-dimensions, m = 1, 2, • ■ • , n — 1, which can be described in the ?i-dimensional region under consideration, and he intro- duced the numbers called after him, which indicate how many cycles of a given class are needed as a basis to represent a general cycle of that class. Attention has already been called to Picard's work on questions in this field relating to algebraic surfaces, § 7. Poincare perceived the value of this branch of geometry for analysis and published a series of papers on the subject.! Fol- lowing Betti, he considered integrals extended over closed m- dimensional manifolds (cycles) in the n-dimensional region, and he found the conditions that the value of the integrals be invariant of a restricted deformation of the manifold; II, § 2. Such integrals may form the basis for determining the Betti numbers. § * Cf. the fragment in his collected works, Werke, 1 ed., p. 448; 2d ed., p. 479. t Annali di mat. (2), 4 (1870-71), p. 140. J Cf., in particular, Journ. Ec. Polytech. (2), Cah. 1 (1895), p. 1; also the account given in Picard et Simart, Fonctions algebriques de deux variables, vol. 1, ch. 2. § An elementary geometric treatment of the analysis situs of hypermanifolds has recently been given by Veblen and Alexander, Annals of Math. (2), 14 (1913), p. 163. LECTURE II SOME GENERAL THEOREMS § 1. Definitions and Elementary Theorems Let F{Zu •'•,Zn) be a complex function of the n complex variables Z/t = Xk -\- iyk, k = 1,2, ■• -yTi, which is defined uniquely at each point of a 2?i-dimensional continuum T. Of the two current definitions mentioned in I, § 1, we will choose the second and say: F is analytic in T if, at every point of T, it admits a derivative with respect to each of the complex arguments Zi, • • • , 2„ and if, furthermore, it is continuous in T. The latter condition turns out to be a con- sequence of the former, cf. § 5, and may, therefore, be stricken from the definition. But it is better to retain it for a time, since it suffices for a simple proof of the integral theorems, and with the aid of these all the principal theorems are readily established. The function F(zi, • • • , Zn) is said to be analytic in a point (ai, • • • , a„) if it is analytic throughout some region T containing the point in its interior. Similarly, F is said to be analytic in a manifold M of one or more dimensions if it is analytic throughout a region T containing M in its interior. If M is closed, i. e,, if M contains its boundary points, then, for F to be analytic in M, it is clearly sufficient that F be analytic in every point of M. The Canchy-Riemann Differential Equations. The differential equations which the real part u (or the coefficient v of the pure imaginary part) of an analytic function satisfies are the following: dxkdxi dykdyi ' dxkdyi dykdx 133 134 THE RLU)ISON COLLOQUIUM. Thus, when n = 2, there are four equations:* -u c-u c~ii^ d-u Cylindrical and B-Regions. A general region of hyperspace can be described analytically by one or more inequalities. Thus the interior of the hypersphere of radius r, with its centre at the origin, is given by the inequality .Tl^ + .T./ + . . . + xj < r\ or by the pair of inequalities : 2 — Vr^ — X-^ — • • • — Xm-\ < Xm 2, the choice is still larger, for the variables may then be divided into two classes, those of one class being transformed projectively, and those of the other class spherically, i. e., so that the Neumann sphere of each variable goes over into a Neumann sphere of a new vari- able. Thus, if z is defined as an algebraic function of x, y by * Osgood, Transactions Amer. Math. Soc, 13 (1912), p. 159. FUNCTIONS OF SEVERAL COMPLEX VARL\BLES. 141 the irreducible equation /(.r, y, z) = 0, it is possible to associate with this function the surface in projective space given by setting Xq Xq _ X3 Xo' It is, however, also possible to put •Tl Xz X = — , y = —, Xq' ^ Xo' Zi -^0 and still again to set _ iEl Vl 2l > y = —, 2 = — Xq yo Zo There is another geometry that is w^ell known, — the geometry of reciprocal radii, or the geometry of inversion. It would, of course, be a proceeding entirely coordinate with that which has been set forth above to extend the finite space of n complex variables to the space of that geometry. These questions could not arise in the case of analytic functions of a single complex variable, for there the infinite region of pro- jective geometry, the geometry of inversion, and the space of analysis are the same, namely, one point. For the case of two complex variables, the infinite region of the space of analysis and the infinite region of projective geometry are different, and moreover the space of analysis and projective space can no longer be transformed on each other in a one-to-one manner and continuously. But the space of analysis is trans- formable in a one-to-one (but non-real) manner, and continuously, on the space of the geometry of inversion. When the number of complex variables exceeds two, all three spaces are distinct.* * For a detailed treatment of these questions cf. a paper by Buchcr, Bulletin Amer. Math. Soc. (2), 20 (1914), p. 185. We note that the infinite region of the space of analysis consists of n complex (n — l)-dimonsional manifolds (hyperplanes) which have as their sole common point the point (00, 00, ..., 00). 142 the madison colloquium. § 4. Rational and Algebraic Functions To Weierstrass is due the theorem that a function of n complex variables which is meromorphic at every point of the space of analysis is a rational function.* Weierstrass did not define the space in which the function is considered. He said " im ganzen Gebiete seiner Verander- lichen." It appears, however, from more explicit statements in similar cases f that he thought of each variable as an arbitrary point of its extended plane. A similar theorem holds for algebraic functions. If a function of n complex variables is finitely multiple-valued and if, in the neighborhood of every point of the space of analysis, the values of the function can be so grouped as to satisfy one or more alge- broid relations, ^ow"* + A^.io'^-^ + h Am = 0, where the ^'s are analytic in the point in question, — and to be exhausted in said neighborhood by these systems, — then the function is algebraic. § 5. Sufficient Conditions that a Function of Several Complex Variables be Analytic In order that a function of two real variables be analytic it is not enough that the function be analytic in each variable sepa- rately when the other is held fast, as is shown by the example: fi^' y) = ,.2 V ,;i » < I ,r I + I ?/ 1 ; '^ ~i y m 0) = 0, the function being considered in the neighborhood of the origin. When, however, we allow the variables to take on complex values, the case stands otherwise. * Journ. filr Malh., 86 (1880), p. 5=Werke 2, p. 129. The theorem was I)iovon by Hurwitz, Journ. fur Math., 95 (1883), p. 201. t Cf. for example Werke, 3, p. 100, 7th Une from end. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 143 Theorem. Let /(.v, y) be defined throughout a cylindrical region (S, S'), § 1. Let/(.r, 6) be analytic in *S, for every choice of h in S'; b, when once chosen, to be held fast. Similarly, let /(a, y) be analytic in S', a being any point of S. Then /(.r, y) is analytic in the two independent variables x, y throughout the region {S, S'). The theorem is readily proven if the further hypothesis be added that the function remain finite, and under this restriction in sufficiently general for many of the cases which arise in prac- tice.* It is, however, of distinct interest to know that the more general theorem is true. This latter result has been established by Hartogs.j The theorem holds for functions of any number of variables. Further theorems of the character of those here considered are given in the next paragraph, Theorems A, B. § 6. Sufficient Conditions that a Function be Rational OR Algebraic Hurwitz's proof of Weierstrass's theorem, § 4, yields more than is contained in the statement of that theorem. By means of it the following theorems can be established. Theorem 1. If /(si, • • • , Zn) is meromorphic at every point of the coordinate axes; i. e., in each of the points (0, •••,0, 2,, 0, •••,0), /.•= 1, ••-,//, where the variable Zk ranges over the whole extended 2A-plane, then f{zi, • • • , Zn) is a rational function of its arguments. This theorem can be stated in the following form. Theorem 1'. If /(zi, • • •, z,i) is meromorphic in each of those points of the infinite region which corresponds to any ?i — 1 north poles combined with any point whatever of the 7ith sphere, then the function is rational. A special case of this theorem is the following. * This theorem was proven bj-^ the author, Math. Ann., 52 (1899), p. 4ll2 t Math. Ann., 62 (1905), p. 1. Cf. also Osgood, ibid., 53 (1900), p. 461. 144 THE MADISOX COLLOQUIUM. Corollary. If f{zi, • • • , z„) is meromorphic in every point of the infinite region of the space of analysis, then / is a rational function of all its arguments. These theorems readily suggest others, in which the word meromorphic is replaced in the hypothesis by analytic; the con- clusion then being that the function is a constant. Theorem A. If /(zi, • • •, Zn) is analytic in every point of the coordinate axes, then / is a constant. The manifold M consisting of the coordinate axes is perfect, and hence / is analytic in a 2?z-dimensional region T enclosing the axes. It is possible, in particular, to choose a positive number h so that / is analytic in the region |zi| < h, " -, \zk-i\ < h, \zk+i\ < h, • • •, |2„| < h, Zk ranging over the whole extended z^-plane; k = 1, • • •, n. Consider / in the region S: \zk\ < h, k = 1, • ' -, n. Let (fli, •••,«„) be a point of this region. The function /(ai, • ••, On-l, Zn) is analytic over the whole extended 2„-plane. Hence it is a con- stant. Hence dZr. = in the point (oi, • • •, «„). But this was any point of 2. It appears, then, that df ^^-0, k==h-.-,n, and from this fact follows the truth of the theorem. As in the case of Theorem 1, so here the theorem admits an alternative statement. Theorem A'. If /(zi, • • •, Zn) is analytic in those points of the infinite region of the space of analysis which correspond to any FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 145 n — 1 north poles combined with any point whatever of the 7ith. sphere, then / is a constant. As a special case of the theorem we have the Corollary. If /(zi, • • • , z„) is analj'tic in every point of the infinite region of the space of analysis, then / is a constant. This last result can be stated in a form wholly independent of any assumption regarding the infinite region. Theorem B. If /(zi, • • •, s„) is analytic at all finite points out- side a fixed hypersphere :* G < .ir + yr + ^'2'+ ••• + yn\ and if / is finite in this region, then / is a constant. Returning now to theorems, relating to rational functions, we have the following. Theorem 2. If f{z\, • ■ • , Zn) is a rational function of each individual variable, when all the others are assigned arbitrary values in the neighborhood of a certain fixed point and then held fast, then / is rational in all its arguments. The proof of Theorem I is covered by Hurwitz's reasoning, and the same is true of Theorem II, provided the additional hypo- thesis is made that the function be analytic in all its arguments in the neighborhood of the fixed point in question. In practice, this further condition appears usually to be fulfilled. For a proof that this condition is a consequence of the others I am indebted to Professor E. E. Levi. Both theorems can be extended to algebraic functions, the hypothesis then being that the function is A^-valued, and that, moreover, it is algebroid, where before it was meromorphic. § 7. On the Associated Radii of Convergence of a Power Series Let be a power series convergent for a set of values of the arguments, * This hypothesis may equally well be written in the form G < \zi\ + ■•• +\Zn\. 146 ' THE MADISON COLLOQUIUM. no one of which is zero.* A set of positive numbers I'l, ■ • -yrn such that the series converges when \xk\ < Tk, k = 1, '-',11, but diverges when \xk\ > Tk, k = \, ' • -, n, is called a set of associated radii of convergence. The numbers ri, • • • , r^ are in general mutually dependent on each other. Thus in the case of the series T,Xl'X2'' = 1 — a*ia:2 it is clear that rir2 = 1. Geometric Interpretation. Geometrically the associated radii of convergence may be interpreted as follows. Denote by Tk the circle \xk\ < Pk and by T the 2r^-dimensional cylindrical region T= (ri, •.., Tn). Let/(a-i, • • •, Xn) be analytic at the origin. Then the p/t's can be so chosen that T lies in the region of definition of the element /(a-i, • • • , a-„) in question. And now let the p/t's increase. Any system of values Pk = fk, k = 1, • • • , n, such that the function is analytic in the corresponding region T = (Ti, ' • ■ , Tn), but no one of the T^'s can be replaced by a larger circle without diminishing some other Tk and have this properly preserved, is a system of associated radii of convergence. Thus we may picture to ourselves a variable cylindrical region T in the domain of definition of the monogenic analytic function /(.Ti, • • •, Xn). Those regions T that reach out to singular points of the function and, moreover, are maximum regions in this * By a convergent multiple series Sw,,i ... ^„ we mean a series such that every simple series formed from its terms converges. If, then, a multiple series converges, it necessarily converges absolutely. Other multiple series have been investigated in recent years by Pringsheim and Hartogs. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 147 sense, that they cannot be expanded for any one of the variables Xk without being contracted for others, yield associated radii of convergence. Detailed Consideration of the Case n=2. The mutual relation's between the r's have been studied extensively. Let the number of variables be two, and let the series be written : (1) T.Cm.nX'^y^ If, for a pair of values a*o, yo, neither of which is zero, the terms of the series (1) remain finite, then it is well known that the series converges (and hence converges absolutely), when I •'^' I < '•. I ^ I < *> where r = \xo\, s = \yo\. Let r have an arbitrary value in the interval < ^ < |a:o|. To this value of r may correspond larger values of 5, — in fact, there may be no limit to s. If, however, the latter is not the case, let 0, the associated radius of convergence is infinite, then it is infinite for every smaller value of r : ^ r ^ ro. As r increases, (p{r) decreases or remains constant; i. e., (p{r) is a decreasing monotonic function of r.f The domain of definition of (p{r) consists of an interval ^ i?o < r < i?i or ^ i?o < /• < ^ , where, however, it is not obvious whether an extremity of the interval shall pertain to the interval or not. The basal theorem relating to (p(r) is the following. Theorem. X Let be a double power series, and let s = (fir), O^Roi;ic at the point in question when a suitable value is assigned to it there.* For example, the function of the single variable z, f(^) = i has an isolated singularity at the point 2=0. But the function of the two complex variables tv = u -{- vi, z = X -\- yi: F{w, 2) = I , has a whole two-dimensional manifold of singularities in the four-dimensional space of these variables, namely, the points {it, ^^ 0, 0). It is a theorem due to Weierstrass and proven by Rungef that to an arbitrary continuum T of the complex s-plane there correspond functions of z which are analytic at every point of T and which furthermore cannot be continued analytically over * This result can be obtained directly from Cauchy's integral formula or Laurent's series. It was stated by Hurwitz in his Zurich address, Verh. des 1. intern. Malh.-Kongr esses, 1897, p. 104. t Ada, 6 (1884), p. 229. 160 FUNCTIONS OF SEVERAL C0:MPLEX VARIABLES. 161 a single boundary point of T. This theorem has, moreover, recently been extended to the most general Riemann's surface.* It is clear from the foregoing ihat such a theorem cannot hold for functions of more than one variable. § 2, XON-ESSENTIAL SINGULARITIES The analogue of a pole of a function of a single variable is a point (ai, • • •, Qn), in whose neighborhood the function can be written in the form U; -f* (.21, ■ • • , 2„) = 777- —:, h\Zi, • • -, Zn) where G and H are both analytic at (ai, • • •, a„), and G(ai, • ", cin) = 0, H{ai, • • • , an) +0. Here, F becomes infinite for all methods of approach to the point, just as in the case n = \. We shall denote such a point as a yole, or as a non-essentially singular 'point of the first kind.] But even a rational function can have a more complicated singularity. Suppose that G and H are polynomials relatively prime to each other, both vanishing at (ai, • • •, a„); e. g., w F{ic,z)=—, {a„a-^) = (0,0). Here, the function can actually take on any arbitrarily assigned value in a point of an arbitrarily assigned neighborhood of the singular point in question. We are led, then, to a second kind of singularity, the function still being of the form (1), but H vanishing also at the point in question, though still being prime to G. Such a point is called a iion-essentially singular point of the second kind. In the neighborhood of such a point, which we will take as lying at the * The question has been treated by Koebe, Freundlich, and Osgood; cf. Osgood, Funktionentheorie, v. 1, 2d ed., 1912, p. 747. t Weierstrass, Werke, 2, p. 156. 162 THE MADISON COLLOQUIUM. origin, (0, • • •, 0), the function can in general* be written in the form: (2) F(zu ■'■,zn)= ^j^^^^j-i_^,,.^Bi "^'^' " • ' "")' where the coefficients A, B are functions of (zi, • • •, 2„_i), each analytic at the point (0, • • • , 0) and vanishing there, the two polynomials in Zn being prime to each other; and where, moreover, 12 is analytic and not zero at the origin. In every neighborhood of a pole there are other poles, their locus being the (2?i — 2) -dimensional analytic manifold or manifolds G(zi, • • •, z„) = 0. But there are no other singularities in the neighborhood in question. For the special case n = 2 the non-essential singu- larities of the second kind are isolated points, since two functions G{iv, z), H{w, z) which are prime to each other, like two poly- nomials having this property, can vanish simultaneously only in isolated points. But when n > 2, there will be a whole (2?i — 4) -dimensional locus of singularities of the second kind, — this locus consisting of a finite number of analytic configurations, each of the dimension in question. In fact, the necessary and sufficient condition that the numerator and the denominator of the fraction in (2) vanish simultaneously is that their resultant vanish. The latter is analytic in zi, • • •, Zn-i and vanishes at the origin; but it does not vanish identically. As regards the poles which lie in the neighborhood of a singu- larity of the second kind, they are situated on the manifold, or manifolds, GiZi, ■ • ', Zn) = 0, and they consist of the totality of such points with the exception of those for which // also vanishes, i. e., the singularities of the second kind. * In any case, a suitable homogeneous linear transformation of zi, • • •, Zn will yield a new function for which the statement is true; cf. IV, § 1. The theorems of the paragraph just cited are assumed in the present paragraph. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 163 A function which has no other singularities in a given region or in the neighborhood of a given point than non-essential ones is said to be meromorphic in the region or in the point. § 3. Essential Singularities An analytic function of a single complex variable z may have an isolated essential singularity, z = a, of either one of two kinds: (a) the function may be analytic throughout the complete neighborhood of the point a except at the point itself, and there neither remain finite nor become infinite; (6) the function may have poles that cluster about the point a, being analytic at all other points of the neighborhood distinct from a. It follows from the first theorem of § 1 that the first case has nothing corresponding to it when we pass to functions of several complex variables. But may not the second case be realized? May not a function of several variables, f{zi, • ■ ■ , Zn), be analytic except for non-essential singularities throughout the whole neighborhood of a point (oi, •••, a„), this point alone being excepted? Weierstrass believed apparently that it can, for he stated the following theorem.* To an arbitrary continuum in the 2/i-dimensional space of the variables (zi, • • • , 2„) there correspond functions analytic or having at most non-essential singularities, but having in every boundary point a singularity of higher order. This theorem, however, is false, as was shown by E. E. Levif in a notable paper published three years ago, to which we shall return later, §§ 8, 9. In particular, it appears that an isolated essential singularity is impossible. § 4. Removable Singularities In his inaugural dissertation RiemannJ stated and proved the theorem whose practical value is so well known, namely, that * Journ.fur Math., 89 (1880), p. o=Werke, 2, p. 129. ^ Ann. di. mat. (3), 17 (1910), p. 61. Levi's paper appeared while a paper of Hartogs, Math. Ann., 70 (1911), p. 217, overlapping to some extent Levi's paper and showing in particular the impossibility of an isolated essential singularity, was in press. t Gottingen Dissertation, 1851, § 12, =Werke, p. 23. 164 THE MADISOX COLLOQUIUM. if a function f(z) is analytic throughout the neighborhood T of a point z = a with the possible exception of this point itself, and if f{z) remains finite in T, then f{z) approaches a limit when z approaches a ; and if the function is defined for z = a as equal to its limiting value there, then it is analytic in this point also. This theorem admits a number of generalizations or extensions for functions of several variables. The most obvious one was stated and proven by Kistler* in the following formulation. Let ^(21, • • • , Zn) be analytic throughout a region T consisting of the neighborhood of a point (oi, •••, a„) with the exception at most of the points of a i^lri — 2)-dimensional anahi;ic manifold L; and let the function remain finite in T. Then the function will approach a limit in the points of L and will be analytic there if suitably defined there. Similarly, Riemann's theorem, that a function f{z) which is analytic in a region S except along a simple regular curve C, where it is continuous, is also analytic in the points of C, can be generalized. If /(21, •••, Zn) be analytic in a 2??-dimensional region except in the points of a single {2n — l)-dimensional analytic! manifold (5, where / is continuous, then / is anah'tic in the points of S also. This theorem is not mentioned by Kistler. A second generalization was given by Kistler, and is as follows. Let f{zi, ■ • • , Zn) be analytic throughout the neighborhood of a point (ai, • • •, (In) with the exception at most of the points of a finite number of analytic manifolds, each of which is at most (271 — 4)-dimensional. No hypothesis, however, is now made regarding the function's remaining finite. Such a function will be analytic in the excepted points also, if properly defined there. For 71 = 2, this becomes the first theorem of § L * Gottingen dissertation, Ueber Funktionen von mehroren komplexen Ver- andcrliclien, § 7, Basel, 1905. This theorem, Hke the original theorem of Rie- mann's, is exceedingly serviceable in practice, and was probably used before Kistler's enunciation and proof of it. An important special case was familiar toWeierstrass; I, § 1, end. A second proof is contained substantially in Har- togs's paper. Math. Ann., 70 (1911), p. 217. t More general manifolds are also admissible. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 165 A special case of this theorem was famihar to Weierstrass, namely, that in which the function/ can be written, in the neigh- borhood of the point in question, as the quotient of two functions, each anal3i;ic and vanishing there; cf. I, § 1, and IV, § 1. This latter theorem of Kistler's admits an extension. The excepted points may fill a (2n — 3) -dimensional manifold, the latter being such that, if we set Zk = Xk + iyk, then three of the 2n coordinates, as yn-i, Xn, yn, can be expressed as single-valued or finitely multiple-valued continuous functions of the remainder in the neighborhood in question. And cases reducible to the latter by linear transformation of the complex variables are obviously included. Related to these theorems more or less closely is a further theorem stated by Kistler,* but not proven by him. From the neighborhood T of a point (oi, • • • , Qn) let the points of a set L be excluded, L consisting of the points of a finite number of analytic manifolds, each of dimension 2n — 4 or lower; and let the remainder of T be denoted by T'. In the region T' f(zi, • • • , Zn) shall be meromorphic. Then the function can have in the points of L no higher singularities than removable and non-essential ones. The proof of this theorem was later given by Hartogs.f We note, however, in closing this paragraph an interesting application which Kistler makes of the latter theorem to a proof of Jacobi's theorem of inversion, I, § 2. § 5. Analytic Continuation by Means of Cauchy's Integral Formula During the last few years a number of important theorems on analytic continuation have been discovered, chiefly through the * L. c. In the light of Gronwall's recent discovery concerning the scope of Cousin's theorem, Eastler's proof was even more restricted than appeared at the time. t Math. Ann., 70 (1911), p. 217. Kistler appears to have had no substantial reason for supposing the theorem to be true, for his proof is based on a mis- understanding of Cousin's results, II, § 9. The chief credit for the theorem would seem, therefore, to be due to Hartogs. 166 THE MADISON COLLOQUIOI, researches of Hartogs and E. E. Levi. We begin with the former. Hartogs' s Theorem* Let B, B' be regular regions of the a:-plane and 2/-plane respectively, and let K be the neighborhood of an interior point yo of B'. Letf{x, y) be a function vAth. the fol- lo'vs'ing properties, cf . Fig. 1 : (a) In the interior of the four-dimensional cylindrical region {B, K), f{x, y) shall be analytic; and, moreover, for every point y' of K,f{x, y'), regarded as a function of x alone, shall be con- tinuous on the boundary C of B. (b) For every point ^ of C, /(^, y), regarded as a function of y alone, shall be analytic witliin B' and continuous on the boundary C of B'. (c) In that part of the boundary of {B, B') which is determined by the points (^, 77) where ^ ranges over C and r} over C',f{^, 77) shall be a continuous function of (^, 77). Then f{x, y) can be continued analytically throughout the interior of the entire cylindrical region {B, B'). The proof of this theorem is simple. For every interior point (a*, 7j) of {B, K),f(x, y) can be represented by Cauchy's integral formula : Again, by Cauchy's integral formula, ,,, , 1 rmn), /(-*) = 2^ ii^''"- Hence where the double integral is extended over the part S of the * Sitzungsber. der Munchener Akad., 36 (1906), p. 223. The formulation here given is slightly different from that of Hartogs. y FUNCTIONS OF SEVERAL COMPLEX VARL\BLES. 167 boundary of {B, B') consisting of the closed surface described in the condition (c) of the theorem. This double integral, however, represents a function which is analytic in the two independent variables (.r, y) throughout the whole interior of (B, B'), and which, furthermore, coincides with the given function throughout the interior of {B, K). Hence the proposition is established. a'-PLANE 2/-PLANE Fig. 1. It is interesting to notice the nature of the hypotheses, (a) imposes a condition on the function in each point of a 4-dimen- sional region; (6) is 3-dimensional, in that it is made up of a 1-dimensional system of 2-dimensional hypotheses; while (c) is 2-dimensional. Again, the points of (a) form a 4-dimensional piece of the 4- dimensional cylindrical region {B, B'). The points of (6) form one or more pieces of the 3-dimensional boundary of {B, B'). This latter manifold, it will be remembered, consists of but a 168 THE MADISON COLLOQUIUM. single piece. Finally, the points of (c) yield one or more 2-di- mensional pieces of the 3-dimensional boundary of (B, B') just referred to, and they also lie in the points of ih). A further aid toward a geometric realization of the hypotheses is obtained if we picture the cylindrical region {B, B') as a rect- angle in the plane of analytic geometry. Here, as in the use of that plane in the study of plane cvu'ves w^hen the complex points are admitted to the discussion, we have, it is true, only a two- dimensional figure for a four-dimensional set of geometric objects; and we have to work by analogy. 1/ •^<^'^^^'t^^y■^■>ky!■^^^;4^■:^■;/■^^■'-^ n l\ tn X Fig. 2. Condition (a) is now seen to refer to the points of a narrow strip that courses the large rectangle Imno, the latter representing the region (5, B'). Conditions (6) and (c) have to do merely with points of the boundary, which lie in the sides lo and mn. In the conclusion, the function is extended over an enlarged region dimensionally coordinate with the slender strip of con- dition (a). The extension of the theorem in the above formulation to the case of w-variables is obvious. For three variables, the geometric interpretation last considered leads to a rectangular parallelepiped, coursed by a slender one with parallel faces, and the further conditions of the theorem are interpretable in terms of regions and curves lying in the faces of the large parallelepiped. Another form of the hypotheses of the theorem, somewhat less general, but more compact, consists in requiring the function j{x\, • • • , a-„) to be analytic FUNCTIONS OF SEVERAL COMPLEX VARL\BLES. 169 (a) in every point (xi, ■ • -, Xk, ff/t+i, •••,«„), where av, i = 1, ' • ' , k, ranges over Bi, and aj, j — k -{- 1, • • •, r?, is a fixed point in Bj'y (b) in every point (^i, • • •, ^k, Xk+i, • • •, Xn), where ^i, i = 1, • • •, k, ranges over the boundary Ci of Bi and Xj, j = ^• + 1, • • ', n, ranges over Bj. The function will then admit analytic continuation throughout the cylindrical region (Bi, • • •, jB„). In the foregoing results is contained the remarkable theorem that a function /(.I'l, • • •, .t„) which is analytic in every boundary point of a cylindrical region {Bi, • • • , Bn) admits analytic con- tinuation throughout the whole region.* This theorem holds for the general case of any four-dimensional region, whether cylindrical or not. Cf. § 9. § 6. Application to the Distribution of Singularities From the main theorem of the last paragraph Hartogs deduces the following theorem relating to the distribution of the singu- larities of an analytic function. Theorem. Let /(.r, y) be analytic in the points (0, y), where 0<\y\< h, and let/ have a singular point at the origin, (0, 0). Then, to each point x' of a certain region B: \ x\ < p, will cor- respond at least one point y' of the region B':\y\ 0, and which has in the former hypersphere a natural boundary. As has already been pointed out, the theorem was stated and proven for cylindrical regions by Hartogs. Theorem 2. This theorem differs from Theorem 1 solely in * Cf. II, § 1. 176 THE MADISON COLLOQUIUM. having the word analytic replaced throughout by the word meromorphic. The second theorem can be proven as follows. We may without loss of generality assume that the boundary of T is pierced by an arbitrary ray from the origin at most in a finite number of points. On each ray which enters T there will, then, be a finite number of segments lying in T. Let AB be such a segment, and let B be the extremity more remote from the origin. Continue /(x, ?/) meromorphically from i? toward ^. If it is possible to reach A on every segment, the theorem is granted. If not, let Q be the first point on AB that cannot be reached from B. Thus, when all segments are considered, a set of points Q lying in the finite region T are obtained, and this set is, from its source, necessarily closed. I^et P be one of its points whose distance from is a maximum. Then, in that part of the neighborhood of P which lies outside of the hypersphere through P with its centre at 0, f{x, y) is meromorphic. The function must, therefore, by Levi's theorem, § 8, admit a meromorphic continuation at P, and here is a contradiction. The first theorem can be proven in a similar manner by the aid of Levi's theorem of § 8, stated for functions required to be analytic instead of being allowed to be meromorphic. It thus appears that an analytic function of two complex variables cannot have a finite lacunary space around which the function is meromorphic. This latter result is in direct contradiction to Weierstrass's theorem of § 3, and appears to be the earliest proof that that theorem is false. From Lemma 2 it follows, however, immedi- ately that an isolated essential singularity is impossible, and thus a more elementary proof is afforded of the incorrectness of that theorem. functions of several complex vael^bles. 177 § 10. Concerning the Boundary of the Domain of Definition of /(.r, y) Let 2 be a simple regular 3-dimensional manifold of 4-dimen- sional space. Then 2 can be represented analytically by the equation on the side of 2 where /(.r, y) is to be mero- morpliic. Denote by (S(c?) the following expression: (do dw ^?i7 + K^^i^yx ^x.^yi) K^x^^Vi ^x^cy^)\^Xydy. dx.dyj' where ^-'=(^.)"+(iy' and where A/V, Ao'V denote similar expressions in yi, y-2. Then must (^{(p) ^0 in all points of 2. If ^ < on the side of 2 where f{x, y) is meromorphic, then must S(^) ^0 in all points of 2. From this result it follows that if there is to exist a function 178 THE MADISON COLLOQUIUM. f\{x, y) meromorphic on one side of S and having S as a natural boundary; and also a function /2(x, y) meromorphic on the other side of 2 and also having S as a natural boundary, then must ^i and having 2 as a natural boundary; — all this, at least, when 2 is suitably re- stricted in extent. § IL A Theorem Relating to Characteristic Surfaces An analytic surface in space of four dimensions may be repre- sented by a pair of equations: (1) w(.Ti, a-2, 2/1, 2/2) = 0, v{xi, X2, 2/1, 2/2) = 0, where u and v are real functions of the four real variables, analytic at the point in question, their Jacobian with respect to two of the variables, — say 2/1, 2/2> — not vanishing there. Levi-Civitaf raises the following question. Suppose two real functions, p and q, are given along such a surface, and are analytic there. Thus p and q may be any functions of .ri, X2 analytic at the point in question, if these are the preferred variables. Does a function of the complex variables exist: iv{x, y), X = x-i. + ix2, y = yi + iy^, * Ann. di Mat. (3), 18 (1911), p. 69. t Rendiconli Accad. Lincei (5), 14 (1905), p. 492. He prefaces his problem by recalling the Cauchy problem for two independent variables, x and y, and an analytic cur\'e C in their plane; an arbitrary sequence of analytic values being assumed along C. FUNCTIONS OF SEVERAL COMPLEX VAKL\BLES. 179 analytic at the point in question and taking on the value p + qi along the surface in question, — all this, at least, in a certain neighborhood of the given point? He finds the answer to be affirmative and the function ic to be uniquely determined, provided the surface is not what he calls a characteristic surface, i. e., a surface along which an analytic function of two complex variables, which is not identically zero, vanishes. In the case of a characteristic surface, there will in general be no solution of the problem. Suppose, for example, that the surface is ?/ = 0, — and the general case of a characteristic surface is reducible to this case. Then w{x, 0) = p + qi, and it is evident that f -\- qi must be a function of x analytic at the given point. If this condition is satisfied, there will be, not a single, but an infinite number of solutions. From these results follow at once the theorems: If /(2*> y) is anal}"tic at a point and vanishes along a non- characteristic siu-face through that point, no matter how re- stricted that surface may be, it vanishes identically. If f(x, y) and 21, • • •, Zn) is any function of Za, z\, • • •, Zn, analj-tic at the origin and vanishing there, but not vanishing identically, it is possible by means of a suitable linear transformation of the n + 1 variables Zq, Zi, • • • , Zn to carry / over into a function Fill', xi, • • ■ , Xn) satisfying the foregoing conditions. Irreducible Factors. On the theorem of factorization can be based a theory of irreducible factors of an analytic function analogous to the theory in the case of polynomials. f First, as regards division. If F{zi, • - • , Zn) and (si, • • • , Zn) are both analytic in the point (a) = (oi, • • •, o„) and <^ does not vanish * Lithographed, Berlin, 1879; Funlctionenlelire, 1886, p. 105 = Werke 2, p. 135. In a foot note of the page last cited Weierstrass says that he has repeatedly given the theorem in his university lectures, beginning with 1860. t AVeierstrass, 1. c. 181 182 THE MADISON COLLOQUIUM. identical ly, but does vanish at (a) ; and if, in the neighorhood of (a), a relation of the form F(Zi, ■■•,Zn) = Q(Zi, •", Z„)*(Zi, • • •, Z„) holds, Q being analytic at (a), then F is said to be divisible by in the point (a). If G(zi, •••, Zn) is analytic in the point (a) = (ai, • • • , ttn) and vanishes there, then G is said to be irre- ducible at (a) if no equation of the form exists: ^(21, • • • , Zn) = Gi(Zi, • • • , Zn) GiiZi, • • ', Zn), where Gi and G2 are both analytic at (a) and both vanish there. Two irreducible factors are equivalent if their quotient, taken either way, presents at most removable singularities. A function G{zi, • • • , Zn) analytic at (a) and vanishing there, but not vanishing identically, can be written in one, and essentially in only one, way as the product of factors each irreducible in (a). A factor which is irreducible at a given point is not necessarily irreducible at every one of its vanishing points which lies in a cer- tain neighborhood of the point. Hence the expression of a func- tion at a given point as a product of factors each irreducible at that point does not always retain this character when that point is replaced by a second root of the function that lies in the neighborhood of that point. The theorem of algebraic geometry that two curves or surfaces which have ever so short an arc or small a region in common, must necessarily have a whole irreducible piece in common, finds its counterpart here. Let F{zi, • • • , Zn) and $(zi, - • -, Zn) both be analytic at the origin and vanish there, and let $ be irreducible there. If F vanishes at all points in the neighborhood of the origin at which $ vanishes, then F is divisible by $. The Roots of an Analytic Function of Several Variables. In the case of analytic functions of a single variable the roots are isolated. This theorem appears to be lost for functions of several variables, since such a function which vanishes at all has an infinite number of roots clustering about any given root. The theorem admits, nevertheless, a perfectly good generalization. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 183 It is not the Individual root, but the monogenic analytic con- figurations which are made up of the roots and which exhaust the latter, that are the analogue of the roots of a function of a single variable. And now it is seen from the factor theorem that the number of such configurations which course the neigh- borhood of a given root is finite. Earlier Sources, As appears from the applications already considered, there are two wholly distinct classes of theorems at issue. The theorem of factorization asserts the existence of an identity in 7i + 1 independent complex variables, the left-hand side being a function F(u; Xi, • • • , .Tn) vanishing at the origin, but such that F{u; 0, • • •, 0) ^ 0; and the right-hand side being the product of the two factors described in detail in the state- ment of the theorem. This theorem is universally admitted to be due to Weierstrass. On the other hand, such a function put equal to : F{u; xi, ---yXn) = 0, defines an implicit function of n arguments. That the latter function is given as the root of a polynomial : where the ^^'s are all analytic in a;i, • • •, Xn at the point in ques- tion and vanish there, follows, it is true, from Weierstrass's theo- rem. But Weierstrass was not the sole discoverer of this theo- rem. The theorem is contained substantially in Cauchy's Turin memoir of 1831.* In that paper, Cauchy showed that, to each point {xi, • ■ • , Xn) lying in a certain neighborhood of the point (ai, ••■, a„) in question, correspond precisely vi roots of the equation F{u; xi, • ", Xn) = 0. Furthermore, if ^(u) be any function of u analytic at the point M = and vanishing there, and if the above m roots be denoted by III, • • • , Um, then the symmetric function $(Wl) + • • • + ^(«m) * Cf. Exercices d'analyse, 2 (1841), p. 65. 184 THE MADISON COLLOQUIUM, is expressed by a definite integral which' is seen to represent a function of Xi, - •• , .t„ analytic at the point (oi, • • •, a„) and vanishing there.* In order, then, to obtain the implicit function theorem it remains merely to set $(w) = u^, k = 1, ' ' - , m, and then express the elementary symmetric functions by the familiar formulas in terms of the Newtonian sums, Furthermore, Cauchy applied his method to the solution of a problem in implicit functions, namely, to the development of a function into a series of Lagrange. Thus this noted series, so prominent in the early history of the theory of functions, again makes contact with modern analysis. There are two other proofs of the implicit function theorem considered above, both of which antedate Weierstrass's publica- gion in the Funktionenlehre, namely, Poincare's and Neumann's, f § 2. A Tentative Generalization of the Theorem of Factorization In the case n = 1, in which F depends on only two variables, u, X, it is possible to dispense with the condition (1) altogether, provided F{ii, x) does not vanish identically, the relation (2) being then modified as follows: F{u, x) = x^ {iC^ + .4i?r-' + • • • + ^„0 Q.{u, x), where Z is a positive integer, or 0. Even the proviso just men- tioned can be avoided if we write F{u, x) = {Aou^ + Aiv^-' + . . . + A„,) U(n, x). * The work is carried through for the case n = 1, our function F{u; Xi, • • • , x„) being represented there by/(a-, y) and the above function ^(u) by F{y). Th(! proof of the theorem of factorization given by Goursat, Cours d'analyse, 2, § 35G, is based on Cauchy's analysis. t Poincar6, Paris, Th6se, 1879, pp. 6, 7. Neumann, Leipziger Berichle, 35 (1883), p. 85; Abelsche Integrale, 2d cd., 1884, p. 125. ^v FUNCTIONS OF SEVEEAL COMPLEX VAEIABLES. 185 Thus we have a form of the factor theorem which holds in all cases and which does not depend on an eventual change of the independent variables by a linear transformation. A corresponding form for the general case, n > 1, would be a valuable contribution, since it is not always feasible, under the conditions of the problem in hand, to make the above linear transformation. The tentative theorem is as follows. Tentative Theorem. Let F{u; xi, •••, Xn) be analytic at the origin and vanish there. Then, throughout a certain neighbor- hood of the origin, T\ \u\ < h, \xk\ < h', k = 1, • • •, n, the following equation holds: F(u; xi, •••, Xn)=(Aou"'+Aiu'^-^-\ f-^m)^(w; xi, ■■-, Xn), where Ak, k = 0, I, ■ • • , n, is analytic in X\, • " , Xn throughout the region | xi \ < h' and vanishes at the origin when Z: > 0; and where 12 is analytic in T and does not vanish there. Aq may or may not vanish. For polynomials the theorem is obvious. I have not succeeded in proving it in the general case except when n = 1. But in my attempts at a proof I have seen nothing that discredits the theorem and much that renders it probable. I tliink the chances are that the theorem is true, and I hope that someone wiU investigate this question. § 3. Algebroid Configurations Consider the function defined by the equation (1) F= u^ + A,ii^~' + . • • + J„. = 0, where Ak{xi, ■••, .i*„) is analytic in the point {x) = (0) and vanishes there, and the polynomial is irreducible. Such a function is called an algebroid function* * Poincare, These, 1879, p. 4. It is sometimes desirable to admit the case that the coefficient of u'" is a function ^o(xi, • • •, x„) analytic at the point (x) = (0) and vanishing there. 186 THE MADISON COLLOQUIUM. Let A(.Ti, • • •, Xn) be the discriminant of F. Then A ^ 0, and to every point (x) in the neighborhood of (x) = (0) in which A 4= there correspond m distinct roots of F. These may be so grouped as to yield m functions Ui, • • • , «m, each analytic in a preassigned point in which A # 0. Moreover, one of these functions can be con- tinued analytically into every other one, and thus they are all elements of one and the same monogenic analytic function. If 71 = 1, we are led to an ordinary Riemann's surface with a single branch point in the point x = 0, in which all m leaves hang together. If w > 1, it is still convenient to think of a Riemannian mani- fold $ of m sheets, or leaves, as we will still say; though these leaves are no longer surfaces, but 2n-dimensional manifolds. We meet here, however, an entirely new order of relations. In the case n = 1, there was but a single branch point. That was fixed, and the junction lines were movable and to a large extent arbitrary. Here, however, the whole locus (2) A(.Ti, -",Xn) = yields points for which two or more of the w^■'s coincide. In such a point, two u's which coincide may or may not belong to functions each analytic at the point in question and satisfying the equation F = 0. In the former case, the Riemann manifold $ has a multiple (2/i — 2)-dimensional manifold, like a multiple point of a plane curve at which all the tangents are distinct and non-vertical, or more generally, at which no two branches are connected with each other. In the latter case, however, we have a whole {2n — 2)- dimensional manifold of branch points, and the corresponding w's are not analytic in (x) at such a point. In other leaves above or below such a point it may, of course, happen that the corre- sponding determinations of u are analytic. There still remain, in addition, the junctions. These are FUNCTIONS OF SEVERAL COMPLEX VAMABLES. 187 (2n — l)-dimensional manifolds, largely arbitrary in location and character, but necessarily passing through the loci of branch points, i. e., the branch manifolds, and along these junctions one branch of the function goes over into another branch, re- maining analytic all the while. A simple example or two will serve to illumine the above relations. Example 1. — u^ — X = 0, the independent variables being two in number, x and y. Here, the space of the independent variables is a four-dimensional real space Ra, corresponding to the tw^o spheres, — the .x-sphere and the ^/-sphere. If we set x = .ti + 1x2, y = y\-\- iy^, the points of i?4 wdll be (.x'l, .T2, y\, 2/2). The Riemann manifold $ is two-leaved. The branch manifold consists of the surfaces: S2': (0,0,2/1,2/2); S2": (00,00,2/1,2/2), where the point (2/1, 2/2) ranges over the whole 2/-sphere. As the junction we may take the 3-dimensional manifold Rz'. (-Ti, 0,2/1, 2/2), ^ a:i ^ CO, i. e., the point (.I'l, X2) is any point of the positive axis of reals, including the points x = and .x = co ; and (2/1, 2/2) ranges in- dependently over the extended 2/-plane. There is wide latitude in the choice of Rz, but it must contain the surfaces S^ and *S2". Example 2. — u~ — xy"^ = 0, the independent variables again being x and y. Here, the two values of w become equal, not only in the points of the above surfaces S2' and S2", but also in the points ®2': (.Ti, x., 0, 0); (B2": (.ri, X2, o), o)). Nevertheless, in the neighborhood of any point of ©2' and ©2" which does not lie on ^2' or So" the values of u can be grouped so as to yield, two functions, each single-valued and analytic through- out the neighborhood in question. The Riemann manifold $ may be taken precisely as before. 14 188 the madison colloquium. § 4. Continuation. The Branch Points of the Discriminant It is important to notice how the dependent variable behaves in the points of a (2n — 2) -dimensional manifold of branch points. If we are at liberty to make, if necessary, a non-singular linear transformation of the x's, we may assume that A(0, ■■•,0,Xn) +0, and hence replace the equation A = by an algebroid equation in Xn. Let (3) D = xj + BixJ-' + ■■■ + Bi = 0, where D is an irreducible factor of A; and let Di{xu ■ • • , Xn-\) be the discriminant of D. Then Z)i ^ 0. For simplicity in the presentation, we confine ourselves to the case that A has no further irreducible factor. Consider a point Pq: (.ri°, • • • .t„_i°) in which Di 4= 0. In the neighborhood of this point the roots of (3) can be grouped to / functions Xn , Xn" , • • • , Xn^^^ each analytic in the above point and all elements of the same monogenic analytic function. If we substitute one of these elements, xj, in the coefficients of (1), the new polynomial, F=u"'+ Aiu"'-' H h Zn = — where Ak{xi, • • -, Xn-i) is analytic in the point (.ri°, • • •, Xn-i^) but does not necessarily vanish there, — will have a common factor with its allied polynomial - dF F' = — . du Consider the greatest common divisor of F and F'. Let its irreducible factors be Gh{u, Xi, • •-, .r„_i), A; = \, • • -yV. In general, v = 1 and 6'i is linear in u. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 189 This case can always be attained by a linear transformation : (4) Xn = Xn + a U, where a is a suitable real positive number. In fact, returning to the arbitrary case of the text, let Pi: {xi^, • • • , Xn-i^) be a point of the neighborhood of Pq in which the discriminant of no Gk vanishes ; let this be true not merely for the particular element Xn that was substituted in F, but for each of the other I — 1 determinations of .t„ given by (3), Xn", • • • , .t„^'\ Finally, let Pi be so chosen that no two 6r's, — whether they belong to the same a*„^'^ or to different ones: .Tn^'^ and Xn^^\ have equal roots. Throughout a certain neighborhood of Pi, then, we have I patches 2i, •••, S^ of the discriminant manifold (3), and the points (a'l, • • •, Xn-i, xj''^), k — I, • • • , I, thus defined in the space of (a'l, • • • , Xn) are the totality of the points of so much of that space as lies in a certain neighborhood of the origin: |av| for granted. But how do we know that it exists? Even for functions of a single complex variable this question, in the general case, was not simple. It is one of the fundamental problems of the theory to show that, to any monogenic analytic function of several complex variables, corresponds a Riemann manifold. One method of attack would be to prove the theorem for a properly restricted algebroid configuration, and then proceed as in the case of functions of a single variable.* Consider so much of the algebroid configuration (1), § 3, as lies in the region | m 1 < A', | .r^ | < lu. The linear connectivity of the corresponding Riemann manifold is not necessarily unity, no matter how far k, hi be restricted. For, in particu- lar, F may be a non-specialized homogeneous polynomial, so that (1) is the equation of a non-specialized algebraic cone of degree m. Parametric Representation im Kleinen. One other theorem we will mention, — the theorem of the parametric representation of an analytic configuration im Kleinen. Let G{zi, • • •, Zn) be a function of Si, • • •, Zn analytic at the origin and vanishing there, but not vanishing identically. Then there exists a finite number of systems of equations (1) Zk = \(xi, • • •, .Tp) = 0, • • •, Dq{xi, ■••,Xp) = 0, where Dk is analytic in the point (x) = (0) and vanishes there, and is irreducible. FUNCTIONS OF SEVEKAL COMPLEX VARIABLES. 195 We can go further and say: To each point (.I'l, • • •, a-p_i) of the neighborhood of the origin, which does not lie on a certain exceptional manifold E, there correspond M distinct points of D, and likewise M distinct points of the neighborhood of (u; x) = (0; 0), for which the equations (1) are satisfied. These points are determined as follows. Xp is given by an equation having no multiple factors : (5) V+ Bix/'-' + . . . + 5v = 0, where Bk(xi, • • •, Xp-y) is analytic at the origin and vanishes there. If {x\, " -, Xp-\) does not lie on E, the roots of (5) are all distinct and analytic in {xi, • • • , Xp-\) and the further functions u\, ' • • yUi which enter to form the roots of (1) are also single- valued and analytic on the analytic configuration or configurations (5) except, at most, for those points for which (a*i, •••, .ip-i) lies on E. The points of E are those whose co-ordinates satisfy at least one of a finite number of equations Ei{xi, • • •, a-p-i) = 0, • • •, Ek{xi, ■ • ', .Tp_i) = 0, where Ek is analytic in the point {xi, • • • , a'p_i) = (0, • • • , 0) and vanishes there, and is irreducible. We can now proceed to treat the points of E in a similar way, and so on. The foregoing formulation is deficient in one respect. In excepting, as the first step, all points whose (.r) belongs to D some points w^ere lost which have not later been regained. Consider the multiply sheeted Riemann manifold corresponding to (4). For a given point of D one point at least of this manifold is to be excluded. But it may happen that points above or below this one, in other sheets, are such that ui and the other li^'s will be analytic there. The number of such systems, {iii, ■ ■ ■ ,ui; i*^i> * • • > ^p) will, however, be less than N. It would be possible to give to this theorem a formulation more closely resembling that of Weierstrass's theorem, § 6. 196 THE MADISOX COLLOQUIUM. Poincare has given a further theorem,* which he regards merely as another form of the theorem of his Lemma IV, cited above. Under the h\'potheses of the last-named theorem he states that the system of equations (3) can be replaced by an equivalent sj'stem: ^k{ui, • • -, ui; xi, • • •, .Tp) = 0, h = I, • ' -,1, in which "^k, in addition to satisfying the conditions imposed on $/.-, is a polynomial in Wi, • • • , uu Special Cases of the Foregoing Theorem. A special case of this theorem has recently been investigated by MacMillan.f It is evident, in the light of ^yeie^strass's theorem, that no one of the functions ^i{ui, - • -, ui; 0, • • •, 0) can vanish identically. Let $i(Mi, • ■ •,iii;0, • • • , 0) be developed into a series of homogeneous polynomials of ascending degrees, and let the polynomial of lowest degree, — the characteristic polynomial, as Mac^NIillan calls it,— be denoted by ip'^'^'^iui, • • •, m), its degree being A\. Mac- Millan considers the case that the resultant R of the characteristic polynomials does not vanish.! Bliss§ has also given a treatment of this case and has obtained the result that, when i? 4= 0, the number .Y has the value: X = n h. 1=1 Another special case of the main theorem, has been investigated by Clements. II Let the Jacobian J vanish in the point (m; x) = (0; 0), and furthermore let Ji = J, Ji = ^7 X = 0, (w; X) = (0; 0); d{ui, U2, ■ • ■, ui) * M6canique celeste, vol. 1, p. 72. t Math. Ann., 72 (1912), p. 157. X Cf. Bliss'.s critique of Poincare's theorem and the results obtained by MacMillan, Transactions Amer. Math. Soc, 13 (1912), p. 135. §L.c. II Bulletin Amer. Math. Soc. (2), 18 (1912), p. 451; Transactions Amer. Math. Soc, 14 (1913), p. 325. FUNCTIONS OF SEVERAL COMPLEX VAMABLES. 197 Jk-i = -.. -^ = 0, (w; x) = (0; 0); a(Wl, ll2, • • ',Ui) Then the hypotheses of the above theorem are fulfilled, and N= L § 8. The Inverse of an Analytic Transformation Let (1) /T„ = fn(:Ui, • • •,Un), where fk{ui, ••■, Un), k = 1, •••, n, is analytic in the point (m) = (0) and vanishes there. If the Jacobian J of the fs does not vanish, it is well known that the equations can be solved uniquely for the u's in terms of the x's, the resulting functions being analytic at the point (x) = (0). To pass to the other extreme, if the Jacobian vanishes iden- tically, there is a relation between the /'s. More precisely, let T: 1 2ik\ < h, k ^ I, • • •, n, be an arbitrary neighborhood of the point (u) = (0) . Then there is a point (a) of this neighborhood and a function F(xi, • • ■ , .t„) which is analytic in {x) at the point {x) = (b), bk = fk{cii, • • • , fln), k = 1, • • • , n, and which has the following property: (2) F{fu ••.,/„) ^0, where (u) is any point of the neighborhood of (a). Thus the n functions /a;(wi, • • •, Un) are connected by an iden- tical relation* (2), The intermediate case, that J vanishes at the point (u) = (0), but does not vanish identically, has been an object of investiga- tion in recent years. * Peano-Genocchi, Calcolo differenziale e integrale, p. 162. Bliss, Prince- ton Colloquium, p. 67, where it is shown that, when ?i = 2, the point (a) may be taken at (0). 19S THE MADISOX COLLOQUIUM. First, let us observe, a general solution of the problem is given by the theorem of § 7 for all transformations (1) which are such that the point (ti) = (0) is the only point in this neighborhood which corresponds to (x) = (0). For the case n = 2 Clements discussed completely the above transformation under this last-named hypothesis. ^loreover, his theorem cited in § 7, and Bliss's results apply to certain classes of transformations of the kind under consideration. Urner* and Dederickf have also studied the problem from a different point of view, — ^that of the effect of the transformation on certain curves which abut on a point where the Jacobian vanishes. Dedericki introduced the determinant J2 (§ 7) in the case / = 2, and Urner extended the definition to the higher J's. * Transactions Armr. Math. Soc, 13 (1912), p. 232. t Ibid., 14 (1913), p. 143. t Harvard Thesis, 1909. LECTURE V THE PRIME FUXCTIOX ON AX ALGEBRAIC COXFIGURATIOX § 1. The Algebrmc Functions of Deficiency 1 and the Doubly Periodic Functions. Generalizations 1. The Riemann's Surface as Fundamental Domain. The algebraic functions of deficiency unity and their integrals are closely allied to the doubly periodic functions and their related functions, the theta and the sigma functions. It is one of the leading ideas which Riemann introduced into the theory and which has been further developed by Klein and his school that these two classes of functions, from a higher point of view, may all be considered as functions on one and the same foundation (Sub- strat), the Riemann's surface, idealized as a fundamental domain. Thus the ?2-leaved surface of deficiency 1 (or, more particu- larly, the two-leaved surface with four branch points) and the parallelogram of periods are, when regarded as fundamental domains on which functions with familiar properties may be defined, equivalent. The Theta Function. The single function in terms of which the group of functions considered in this theory can be expressed is, when w^e make the parallelogram of periods and its congruent repetitions the domain of the independent variable, the theta or the sigma function. The characteristic properties of this func- tion are: (a) that it is single-valued and analytic within and on the boundary of the parallelogram; (6) that its values in corresponding points of the boundary are related to each other in a simple manner, namely, a{n -\- (jo) = — e ^ '^cr{u), g{u + w ) = — c ^ - ^(r{u) ; 199 200 THE MADISON COLLOQUIUM. (c) that it has a single root of the first order in the parallelo- gram. These properties can be followed with ease when the function is transplanted to the two-leaved (or w-leaved) algebraic surface F, spread out over the z-plane. If a system of cuts is made in F so that a simply connected surface F' with a boundary is generated, a branch of the theta function will be single-valued in F'. This branch will be analytic in all the ordinary points of the surface and continuous in the branch-points, and its values on opposite sides of a cut will differ from each other by a factor always finite and dift'erent from zero. The point oo plays no exceptional role. Finally, the branch in question will have a single zero of the first order in F'. If the function is considered on F, it will be infinitely multiple- valued. But in the neighborhood of any point its values can be grouped to branches each single-valued there, analytic in the ordinary points, and continuous in a branch point. The Independent Variable. I spoke above of the single function in terms of which the group of functions considered in this theory can be expressed. But a function implies an independent, as well as a dependent, variable, and the theta function in the ordi- nary, restricted, sense is simple because of a felicitous choice of the independent variable. If we follow this variable over the fundamental domain taken now as the w-leaved surface F', we find in it a function on this domain, (a) which is everywhere analytic in the ordinary points and continuous in the branch-points and at infinity; (6) which takes on boundary values differing by an additive constant across a cut; and (c) which maps the neighborhood of any point, — even though this be a branch point, — on a smooth single-leaved patch in the other plane. — The function happens to be in this case the every- where finite integral of the algebraic configuration. Generalizations for p> I. On an n-leaved algebraic surface of deficiency p > I the algebraic functions and their integrals FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 201 present themselves without difficulty. But they do not yield a transition to a new fundamental domain on which a function wdth the essential properties of the theta function is readily defined. Riemann constructed functions in a measure akin to the elliptic thetas by means of the theta functions of p arguments. But, aside from the fact that his functions have in general p roots, — never a single simple root, — they may in particular vanish identically. Weierstrass, on the other hand, introduced functions which are single-valued and in general analytic on F, but which have a finite number of essential singularities. A way out was found by Klein in the use of homogeneous variables, already employed by Aronhold and Clebsch in the study of transcendental functions.* Klein perceived still greater possibilities in these ideas and carried through the definition of a function which, considered on the allied manifold in the space of the homogeneous variables, — aUied, I mean, to the Riemann's surface, F, — is a generalization of the elliptic theta function, namely, his yrime function'^ 12(.i-i, .r2; yi, 1/2). The latter is a function, not of two variables x, y corresponding to two points of the given algebraic configuration, one of which, y, may be thought of as a parameter, x being the variable; but of four independent variables xi, X2, yi, y^- It is homogeneous in Xi, x-2, and also in 2/1, ?/2. In Klein's investigations there are two very distinct things which he desires to accomplish. He wishes, it is true, to find a generalization of the elliptic theta function. But he also wishes to obtain a function which will formally be invariant of certain hnear transformations, — often the collineations of the space in which the basal algebraic configuration (Grundkurve) is inter- preted. To accomplish the latter end, the value of the homogeneous * This method was expounded systematically in Clebsch and Gordon's Abelsche Funktionen, of the year 1866. tGottingen lectures on the AbeUan functions, S.-S., 1888, to S.-S., 1889; Math. Ann., 36 (1889), p. 1. 202 THE MADISON COLLOQUIUM. variables he employs is unquestioned. But in so proceeding, the former object is made secondary, — at least, the homogeneous variables must be accepted from the outset, and he does not obtain in the end a function of a single variable and a single parameter, like the elliptic theta transplanted to the surface F. I propose to give here a direct solution of the former problem. What has all this to do with functions of several complex variables? Just this, that the methods of that theory yield proofs where proofs, in the theory as developed by Klein, are lacking. One word as to the importance of this mode of treatment. The algebraic functions and their integrals occupy a central position in analysis through their relation to the geometry of algebraic curves and surfaces, the theory of linear differential equations of the second order with algebraic coefficients, and the automorphic functions of one and of several variables, including the periodic functions of several variables. The progress of mathematics in the future, even more than in the past, will de- pend on the rapidity with which the recruits can be despatched to the frontier. As a result of the theorems of uniformization now rigorously established an improved treatment of the alge- braic functions and their integrals has become possible and, by reason of its simplicity and generality, appears suited to super- cede the methods hitherto used. In this treatment, the prime function as developed in the following paragraphs is the domi- nating factor and may be made the basis for the whole theory. § 2. The Prime Function Defined as a Limit Generalizing from the elliptic case considered by Aronhold and the hyperelliptic case, which he himself had treated at length, Klein introduced, for an arbitrary algebraic configura- tion, reduced by birational transformation to a normal form, an expression which he called an " everywhere finite dift'erential," and which he writes as dux- It is sufficient for our present purposes to know that this expression is analogous, for the neigh- FUNCTIONS OF SEVERAL COMPLEX VARL\BLES. 203 borhood of a given point of the configuration, to what would appear in Weierstrass's theory as x(i)dt, where t denotes the parameter by means of which the neighborhood in question is uniformized, and x(0 is analytic and does not vanish there. Let P^r,{^) be an Abelian integral of the third kind with its logarithmic discontinuities in the points x = ^, x = rj, and let INIoreover, P^y,ix) shall be so chosen that -^ fT, -^ xy' Klein defines his prime function fi(.ri, X2; yi, yi) as the follow- ins: limit: do^xdooye ''•^ dx=0, dy=Q We can now state the definition of the prime function which we propose to develop in detail in the following paragraphs. Let the algebraic configuration be an arbitrary one of deficiency p > 1, and let it be uniformized by automorphic functions with limiting circle in the /-plane.* Let the integral P, transferred to the /-plane, be wTitten Then (1) fi(/, r) = Km yJAtAre ''^ At=0, At=0 In form, then, the definition is identical with Klein's.! But whereas Klein's dcox is single-valued on a homogeneous configura- tion corresponding to the given algebraic configuration, our dt is not invariant of the transformations of the automorphic group. Transferred to the surface F it is infinitely multiple-valued. On the other hand, Q(t, r) is a function of the two independent variables t, r, each being chosen arbitrarily in the fundamental * The details of this uniformization are set forth in the second edition of the author's Funktionentheorie, vol. 1, 1912, ch. 14. t Math. Ann., 36 (1889), p. 12. Cf . also Klein's account of the relation of his prime function to Weierstrass's E{x, y) and Schottky's E{^, rj); ibid., p. 13. 15 204 THE MADISON COLLOQUIUM. domain corresponding to the given algebraic configuration, and not on an allied configuration in the space of the homogeneous variables. Here, r plays the role of a parameter, t being the (single) independent variable. It is, however, important to know the nature of the depen- dence of 12 on both arguments, regarded as independent variables. So far as the analytic character of the dependence is concerned, theorems in the newer theory of functions of several complex variables afford precisely the tools that are needed. § 3. The Existence Theoreais It is evident that, if we are to deal with such a limit as the one here considered and infer the analytic properties of the limiting function, it will not suffice to study the function P|^ or Pf/ merely in its dependence on one variable at a time. The theorem of II, § 5, combined with the existence theorems as developed by Neumann, enables us to obtain with ease the foundation needed. Let us consider first, as lying nearest to the theory of Neumann,* an arbitrary algebraic Riemann's surface F and two points ^, -q of the same. Moreover, ^, -q shall be ordinary points, and it shall be possible to enclose them within a circle K not including any branch-point in its interior or on its boundary. Let ^, 7} be joined by a right line L, and let F be cut along L. Then there exists a logarithmic potential function v, single- valued and finite in the severed surface, harmonic at all ordinary points and continuous in the branch-points and at infinity, admitting harmonic continuation across L from either side, and such that Let d be defined in the region consisting of K cut open along L as follows: d — arc (s — ^) — arc (s — -q) = tan ^ rr — tan ^ r , X — t; X — rj - T < d < IT, * Abelsche Integrale, 2d ed., 1884, chs. 16-18. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 205 where z = .t + yi, ^'= ^' + ^"i, v ^ v' = v' + v"i- Then (2) t^= d+f(x,2j), where /(a-, y) is harmonic throughout K. By means of this function: v = v{x,y;^\^";v',v"), 2p everywhere finite logarithmic potentials can be constructed^ each admitting the modulus of periodicity 1 across one of the 2p cross cuts, but being single-valued across each of the others. In fact, let C be a loop cut not passing through a branch point. Mark on C n points ^k, Jc = 1, • • •, n, so chosen that about two consecutive points a circle K can be drawn. The function v being formed for each pair of consecutive points, the sum of these n functions will be an everywhere finite logarithmic poten- tial with modulus of periodicity 2t across C. These 22? functions are seen to be linearly independent. Out of them a normal system of p everywhere finite integrals can now be constructed: (3) Ax, A2 • •• Ap B, •• • Bp Wi Tri • •• an •• ' aip U'2 iri •• ^21 • • «2p Wp • iri flpi • • • app where au = aa. Furthermore, if we denote the conjugate of the above function v by — u, then u-]r vi is an integral of the third kind. Such an integral can be obtained for an arbitrary pair of ordinary points ^, 77 of the surface by joining these points by a curve L, interpolating on L n — 1 points so that two consecutive points lie in a circle K, forming the foregoing function for each pair of consecutive points, and adding these n functions together. 206 THE MADISON COLLOQUIUM. Finally, this integral can be reduced to a normal integral n|,(z) with vanishing moduli of periodicity across the A cuts: (4) n.Jz) Ai Bi B. 2ivt> ' .•• 2^!" •• where iOk(z) denotes a branch of the function taken in the simply connected surface F', and (5) wt = iVkiO — Wk(v)- The integral U^ni^) is completely determined save as to an additive constant, which is any function of |, rj. If we set (6) ni;" = n^,(2) - u^,(io), then* (7) n|; = nf:^. The scheme of the moduli of periodicity of the function 11^^, when regarded as a function of one variable at a time, is as follows. f A, . • Ap Bi • Bp X . • 2ui^ • • 2wp y • . -2tvp ' • • - 2wp ^ • . 2w7 • • 2tv;' V • • -22V? '• ■ - 2wl' (8) In a similar manner, the normal integral of the second kind is obtained :| (9) Z^iz) = ^-^ + mz), where ^ is an ordinary point and 21(2) is anal}i;ic at s = ^. (10) ZM Ai Br Br - 2^k{z) = I be used. Any other integral of the third kind, PJt', which permits the inter- change of parameters and arguments would serve equally well. Thus a prime function flp(^ t) would result which has the same properties (a), (6), (c), but for which (di) is replaced by a pair of similar equations, each right hand side having an exponential factor whose exponent is a linear function of ti'l''-, • • • , Wp*-'. 216 THE MADISON COLLOQUIUM. be any transformation of G. Then W, r)={t- r') Q'{t, r), where Q,'{t, r) is anal}i;ic in {K) and Q'{i, t) + 0. m., r) = Vl: (t) Q(t, t) ; (a) 1 , Here, each square root denotes a function of t analytic in K and different from zero there. The vahie of these functions is given below. Furthermore id') Q(t,T^) = and let it be held fast. Form the function ^g t) m, r) ' Then this function, regarded as a function of t, will be analytic in K except for removable singularities in the points t = r and in the images of this point under the group G. Let it be defined in these points as equal to its limit. The new function is analytic without exception in K, and does not vanish there. From {d) it follows further that this function is in\'ariant of the transformations of G. It is, therefore, a constant, as can be seen at once by transforming it to the ?i-leaved surface F. Hence -^ = fir), nt, T) = fir) Qit, r), fir) + 0. This last relation is an identity in t, r, and hence can equally well be written in the form ■^ir, t) = fit) fi(T, t). Now apply the property represented by (6). It follows that -■^it,r)= - fit) Qit, r). Hence fit) = fir), and this completes the proof. From the foregoing result it is seen that the properties (a), • • • , id) can serve as the basis for an independent definition of the prime function Qit, r). Thus the function might be represented by an infinite product, as Weierstrass defined his elliptic cr-f unction. And just as AYeierstrass made the latter the basal function for the whole theory of the elliptic functions, so the algebraic functions of deficiency p > 1, and their integrals, can be represented in terms of Qit, r). We proceed to give the fundamental formulas. 218 THE MADISOX COLLOQUIUM. § 9. The Abelian Integrals in Terms of the Prime Function* The Functions 11^,. (0, H^V- From (c) and {d) of § 7 it is seen that the formula ^{t, a) n..(0 = log m, r) gives a particular normal integral of the third kind, the general integral differing from the above by an additive term which is an arbitrary function of a, r. In the absence of any reason to the contrary we set this term equal to zero, i. e., we lay down arbitrarily the definition 1. Thus this integral is completely determined. If we set U = L^{t), tp+a = Lp+a{t), a = 1, P, dt ^«' dt P+a dt t P+a> we find n..^(^) = n,,(o + 1 log (rl, U,^^^,M = n,,(/) + i log(r;^„ + 2iC - a,.', n. .. , (0 = n,,(o - i log t;+, - 2u'r + a. II The function n^'. is now represented as follows: Q(s,a)n(t,T) n^, = log 0(5, r) Q{t, a) ' For this function, regarded as a function of s alone, the formulas (yli)^hold without modification. When t is the sole independent * The deductions of this paragraph are suggested by corresponding formulas in the case p = 1, the function il{t, t) corresponding to d{t — t) or ff{t — t), Cf. Klein, Math. Ann. 36 (1889), p. 11. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 219 variable, the second term on the right of the second formula in (^i) is reversed in sign. On the other hand, the formulas {A2) are simplified. (^2') n ' = n ' n;;^^, , = n;; + 2<' The new formulas (As) are similar, the last term being reversed in sign. The Functions w'J, $a(0- From {A^') and II. we obtain: III. or ,/' = 1 ^{s, Cp+„) 12(f, c) w„ = ^ log f^-:x7^7:r^: — t = 2 log -7^77^ + const. fi(f, c)fi(a, Cp+„) ^{t, c) The expression on the right-hand side is multiple-valued, but the different values can be grouped so as to yield single-valued functions each analytic in K. We choose that one of these functions which vanishes when t = s, ot t = a. A second expression for w'J as the integral of a single- valued function is given below. Formula VII. Let ^Jf) be defined as follows: ^a{t) = dt Then IV. $.(0 = 1^^ log ^{t, Cp+^) n(t, c) ' (t) dt. The Functions Y^it), Y'l', Let V. F,(0 = ^ u^rit) = - |: log n{t, r) = ^-^ + m, t). If we differentiate (^1) with respect to r, we obtain I ^'^^"^ ^ ^'^^^' 16 220 THE aiADISOX COLLOQUItTM. If, furthermore, we differentiate (^3) with respect to r, we have (^2) 'p+a 'p-ra. T, ■p-ra' p-\-a where the accents denote differentiation. Let Ft' be defined as follows: ir = Y^(s) - Y^it). For this function, regarded as a function of s, formulas (£1) hold unchanged. TMien t is taken as the sole variable, the last term in these formulas is reversed in sign. On the other hand, (^2) is replaced by (B,') ytt __ y»« yt __ ^ yst From (Bi) it follows that VL ^a{T) = ^{YM- F,(W«)}. VII. iv'J=hf{YM- i;(c^JWr. The Derivatives of ^{t, r). Let "i(^> r) = %{t, t) = dQ{t,T) dt ' From V. it follows that VIII. 9.,{t,r) = - Qit,r)Y,(t). From (h) we have: Qi{t, t) = - Qiir, t). Hence IX. ^lit, r) 9it,r)Y,(T). FUNCTIONS OF SEVERAL COMPLEX VARLiBLES. 221 The Functions Y'~^\t). From the relations dr (t- ^)2-t--iil^. ^>'> d--'Y.(t) On -1)1 where %m-i(t, t) is analytic within and on the boundary of (^5, %), we are led to define Y^:^''\t) as follows: ■' (m — 1) ! c5t"* 1 Here, From (5i) we obtain the formulas: c yrco = YT\t), iC) YT\tv+.) = Y^rXt) - (,„ ! i^; e-^Xr). 10. The Integral of the Second Kind on F The function Y^{t), when transferred to the ?i-leaved surface F, is an integral of the second kind with a simple pole and with its moduli of periodicity corresponding to the yl-cuts all zero. It differs, however, from the integral there taken as the normal integral, namely Z^(z), as follows. We denoted by Z = (p{t) the function which maps the n-leaved bounded surface F' on the fundamental domain %. Let r be a point of % corresponding to an ordinarj^ point ^ of F'. Then ^^(^) = ^4-^ + 21(2), YAt) = ^ + m), where 2t(2) and 5B(^) are analytic respectively in the points z = ^ and t = T. 222 THE M.iDISON COLLOQIHUM. It follows, then, since Z-^= cp'(r)(t - r) + -^ (t - rf + 91 and W — Zl Wl Let (1) be uniformized as above by the function z = 1, there is the additional circumstance that both R and S are doubly connected regions. The number of independent variables of the allied homo- geneous configuration is 3. To uniformize the configuration which corresponds to {A) let ^{t, Ci) • • • ^{t, ci) ^. be the least common multiple of the denominators of z and w in (2). Then we set .ri = p£l{t, ci) '• • ^{t, ci), and X2, Xz are given by multiplying p by a suitable product of fl- factors. A fundamental domain is here furnished by the cylindrical region (R, %), corresponding to the points (p, /), where < | p | < qo and t lies in g. In case the product that represents Xi contains factors not appearing in the product for Zi, the number-pairs (zi, Zo) will not stand in a one-to-one and continuous relation to the number- pairs (a:i, .To), since the latter will contain the number-pair (0, 0), — a fact of importance in the theory of homogeneous variables. Such uniformizations as the foregoing are of use in studying 228 THE MADISON COLLOQUIUM. the configurations in the space of the homogeneous variables which are alHed to a given algebraic configuration. § 14. LiXEAK DiFFEKENTIAL EQUATIONS ON AN ALGEBRAIC Configuration, and the Factor (p'{t) In his further development of Riemann's programme relating to the determination of linear differential equations by their monodromic group Klein has studied differential equations of the type (A) ^ + P{u, z)^-\- Q{u, z)U=0, where the coefficients P and Q are single-valued functions on a given algebraic Riemann's surface F corresponding to the irre- ducible algebraic equation /(?f, 2) = of arbitrary deficiency p; the coefficients having no other singularities on F than poles, and being, therefore, rational in ii, z. Let P and Q be further so restricted that the singular points of (A) are all regular, and let p > 1. Among such differential equations the subclass is of especial interest whose members have no singular points what- ever. If Ui and U-z be two linearly independent solutions of such an equation and we set* (1) g=. then the neighborhood of an arbitrary point of F is mapped by this function on the smooth neighborhood of a corresponding point of the extended s-plane (or sphere) . The function s is multiple-valued on F. When z describes a closed path on F, a given determination of s, continued analyti- cally along this path, goes over into a linear function of the initial determination : * The notation s was used by Schwarz in similar cases, to whom are due the oarliest investigations in this field which appeared after Riemann's fundamental memoirs. Klein uses the letter ij in this sense. FUNCTIONS OF SEVERAL COMPLEX VARIABLES. 229 ys -{- Thus s is homomorphic, § 5. The present class of equations contains dp — 3 parameters, and it is a problem conceived in the spirit of Riemann's theory to find a vital property of the solutions which shall suflBce com- pletely to determine these parameters and thus single out from the class a unique member. Klein* has given the following solution of this problem. Let F be rendered simply connected by a system of cuts, the bounded surface being denoted by F'. Then the function (1) will map F' on a simply connected region of the 5-plane. This region will have no branch points; but it may overlap itself, and even if this were not the case, the further regions obtained by allowing z to cross the boundary of F' and then describe F' again may conceivably overlap one another. Let the totality of such regions be denoted by S. As a first restriction on the present class of differential equations Klein demands that 2 shall be simple, i. e., consist of a single- sheeted region. To state the requirement in another form, it is this. The function s is multiple-valued on the closed surface F. And now we demand that the values which s takes on in a given point of F shall all be distinct, no matter where this point be chosen. We arrive, then, at a class of differential equations among those under consideration whose allied function s is such that, by means of it, the algebraic configuration f(u, z) = can be uniformized. The functions of s that here present themselves, namely u and z, are single-valued automorphic functions. But the differential equation (A) is still not uniquely determined. The final requirement is this. The region 2 shall consist of the interior of a circle. It is still possible to pass from one circle in *Math. Ann., 19 (1882), p. 565; 20 (1882), p. 49; 21 (1883), p. 141. Cf. also Illein's Gottingen lectures, Ueber linearo Differentialgleichungen zweiter Ordnung, 1894 (lithographed). 230 THE MADISON COLLOQUIUM. the 5-plane to any other by a linear transformation of .9. If, however, we regard all such differential equations (A) as equiva- lent, — their Schwarzian resolvent [s]g= R{u, z) will in fact be the same for all, — we have the result that the differential equation {A) is uniquely determined by the above requirements.* Thus it appears that, when an algebraic function of deficiency 2? > 1 is given, a differential equation corresponding to it can be so chosen that s is precisely the function which we obtained by conformal mapping as t, namely (13) in § 5. From the foregoing developments we conclude that the Abelian integrals corresponding to F, when considered in their dependence on their parameters, form a class of functions which, in important respects, is incomplete. The factor (p'(t) = ll{dildz) is an essential accessory, and this constituent is supplied by the linear differential equation (A), whose parameters are determined in the spirit of Riemann and from a point of view similar to that which has dominated a long line of important researches in another branch of modern analysis, — I refer to the theorems of oscillation of Sturm and Klein. * The demand that 2 be a circle is not the only one which leads to famiUar functions. Thus we might have demanded that the boundary of 2 consist of a discrete set of points, — discrete, as this term is defined in the author's paper in the Annals of Math. (2), 14 (1913), p. 143. We should then have been led to the automorphic functions of the Schottky type. Again, the differential equation (A) would have been uniquely determined. c ^/ L UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. 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