key: cord-0224276-7fdez42q authors: Matsusaka, Toshiki; Ueki, Jun title: Modular knots, automorphic forms, and the Rademacher symbols for triangle groups date: 2021-09-02 journal: nan DOI: nan sha: 7b4eed6caf591984b110aa12d295fa4740b75417 doc_id: 224276 cord_uid: 7fdez42q '{E}.~Ghys proved that the linking numbers of modular knots and the"missing"trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${rm SL}_2mathbb{Z}$. In this paper, we replace ${rm SL}_2mathbb{Z}=Gamma_{2,3}$ by the triangle group $Gamma_{p,q}$ for any coprime pair $(p,q)$ of integers with $2leq p 2 and c > 0 so that we have M −1 γ γM γ = ξγ 0 0 ξ −1 γ with ξ γ > 1 for some M γ ∈ SL 2 R. Then the corresponding modular knot C γ is explicitly defined by the curve C γ (t) = M γ e t 0 0 e −t , 0 t log ξ γ . Note in addition that the unique cusp orbit of the unit tangent bundle T 1 PSL 2 Z\H is the missing trefoil K 2,3 in S 3 . The discriminant function ∆ 2,3 (z) = q ∞ n=1 (1 − q n ) 24 with q = e 2πiz and z ∈ H is a well known modular form of weight 12. The Rademacher symbol ψ 2,3 : SL 2 Z → Z is defined as the function satisfying the transformation law log ∆ 2,3 (γz) − log ∆ 2,3 (z) = 12 log(cz + d) + 2πiψ 2,3 (γ) for every γ = a b c d ∈ SL 2 Z and z ∈ H. Here, we take branches of the logarithms so that we have Im log(cz + d) ∈ [−π, π) and log ∆ 2,3 (z) = 2πiz − 24 ∞ n=1 d|n d −1 q n . In [Ghy07, Sections 3.2-3.3], Ghys precisely asserts the following; Let γ = a b c d ∈ SL 2 Z be a primitive element with tr γ > 2 and c > 0. Then the linking number is given by lk(C γ , K 2,3 ) = ψ 2,3 (γ). The original Rademacher symbol Ψ 2,3 : SL 2 Z → Z is a class-invariant function initially introduced by Rademacher in his study of Dedekind sums ([Rad56] , see also [RG72] ). This function is highly ubiquitous, as Atiyah [Ati87, Theorem 5 .60] proved the (partial) equivalences of seven very distinct definitions and Ghys went further. Ghys indeed worked in his 1st proof with the slightly modified function ψ 2,3 : SL 2 Z → Z, which is not a class-invariant function but coincides with the original symbol Ψ 2,3 at every γ ∈ SL 2 Z with tr γ > 0. In this paper, we proactively take advantage of Ghys's 1st treatment and afterwards discuss the original symbol. Ghys's theorem for (2, 3) generalizes to any (p, q) in two directions; Let SL 2 R denote the universal covering group of SL 2 R and Γ p,q < SL 2 R the inverse image of Γ p,q . Let G r denote the kernel of a surjective homomorphism Γ p,q ։ Z/rZ, which is unique up to multiplication by units in Z/rZ. Then we have the following. -[Tsa13] The spaces Γ p,q \ SL 2 R ∼ = Γ p,q \ SL 2 R are homeomorphic to the exterior of a knot K p,q in the lens space L(r, p − 1), where K p,q is the image of a (p, q)-torus knot K p,q via the Z/rZ-cover h : S 3 ։ L(r, p − 1). -[RV81] The space G r \ SL 2 R is homeomorphic to the exterior of the torus knot K p,q in S 3 . Modular knots in L(r, p − 1) − K p,q ∼ = Γ p,q \ SL 2 R are defined in a similar manner to the case for SL 2 Z, so that each of them corresponds to a conjugacy class of primitive hyperbolic elements of Γ p,q . In order to define the Rademacher symbol ψ p,q for the triangle group Γ p,q , we invoke the theory of harmonic Maass forms for Fuchsian groups; we construct a harmonic Maass form E (p,q), * 2 (z) and a mock modular form E (p,q) 2 (z) of weight 2 to define a suitable holomorphic cusp form ∆ p,q (z) of weight 2pq, that is, ∆ p,q (z) has no poles and zeros on H and has a unique zero of order r at the cusp i∞. The Rademacher symbol ψ p,q : Γ p,q → Z is then defined as a unique function satisfying the transformation law log ∆ p,q (γz) − log ∆ p,q (z) = 2pq log(cz + d) + 2πiψ p,q (γ) for every γ = a b c d ∈ Γ p,q and z ∈ H under suitable choices of branches of the logarithms. Among other things, we prove the following assertion in Section 4. Theorem A. (I) Let the notation be as above. Let γ = a b c d ∈ Γ p,q be a primitive element with tr γ > 2 and c > 0. Then the linking number of the modular knot C γ and the image of the missing torus knot K p,q in L(r, p − 1) is given by the Rademacher symbol ψ p,q (γ) as lk(C γ , K p,q ) = 1 r ψ p,q (γ) ∈ 1 r Z. (II) In addition, let C ′ γ be a connected component of the preimage h −1 (C γ ) via the Z/rZ-cover h : S 3 − K p,q ։ L(r, p − 1) − K p,q . Then the linking number is given by In the course of the proof, we obtain the following as well. Theorem B. The Rademacher symbol ψ p,q : Γ p,q → Z has the following five (partial) characterizations. (1) Definition (Subsection 2.4): log ∆ p,q (γz) − log ∆ p,q (z) = 2pq log(cz + d) + 2πiψ p,q (γ). (2) Cycle integral (Subsection 3.1): Let γ ∈ Γ p,q be as in Theorem A. For the harmonic Maass (3) 2-cocycle (Subsection 3.2): Define a bounded 2-cocycle W : is a unique function satisfying 2pqW = −δ 1 ψ p,q . (4) Additive character (Subection 3.3): Let χ p,q : Γ p,q → Z denote the additive character defined by χ p,q ( S p ) = −q and χ p,q ( U q ) = −p. Then ψ p,q (γ) = χ p,q ( γ) holds for any γ ∈ Γ p,q and its standard liftγ. (5) Linking number (Subsection 4.2): Theorem A (I). The first two properties are described by means of modular forms, while the third and forth are related to the universal covering group. The final property is of low dimensional topology. Although ψ p,q is not a class-invariant function, it seems to be rather a natural object in some aspects and easier to treat, as (1), (3), (4) explain. Besides, we may modify ψ p,q to define several class-invariant functions, namely, the original Rademacher symbol Ψ p,q , the homogeneous Rademacher symbol Ψ h p,q , and the modified Rademacher symbol Ψ e p,q with distinct advantages. Our results mainly concern Ghys's first proof and briefly a half of his second proof. Those are comparable with the results of Dehornoy-Pinsky [Deh15, DP18] on templates and codings related to Ghys's third proof (cf. Subsection 5.1). We remark that (mock) modular forms for triangle groups are quite less studied than those for congruence subgroups of SL 2 Z, although they would also be of arithmetic interest; For instance, Wolfart [Wol83] showed that Fourier coefficients of holomorphic modular forms for the triangle group are mostly transcendental numbers (see also [DGMS13] ). Our study would hopefully give a new cliff in this direction. The rest of the article is organized as follows. In Section 2, we recollect harmonic Maass form for the triangle groups to construct the harmonic Maass form E (p,q), * 2 (z) and the holomorphic cusp form ∆ p,q (z). In Section 3, we define the Rademacher symbol ψ p,q for Γ p,q and prove the equivalence of (1)-(4) in Theorem B. In addition, we discuss several variants of ψ p,q . In Section 4, based on Tsanov's group theoretic study, we establish Theorem A on the linking numbers of modular knots in L(r, p − 1) and S 3 , completing a generalization of Ghys's first proof. Further more, we define knots corresponding to elliptic and parabolic elements to extend the theorem and give a characterization of the modified symbol Ψ e p,q via an Euler cocycle, to justify Ghys's outlined second proof. Finally in Section 5, we give remarks on templates and codings and on the Sarnak-Mozzochi distribution theorem for Γ p,q , and attach further problems. In this section, we introduce harmonic Maass forms for a triangle group Γ p,q . In particular, we construct two important functions E (p,q), * 2 (z) and ∆ p,q (z). The function E (p,q), * 2 (z) is a unique harmonic Maass form of weight 2 on Γ p,q with polynomial growth at cusps, and the function ∆ p,q (z) is a unique holomorphic cusp form of weight 2pq with no poles and zeros on H. Let (p, q) be a coprime pair of integers with 2 p < q and put r = pq − p − q as before. In this subsection, we define the triangle group Γ p,q as a subset of SL 2 R and recall several properties. Recall that PSL 2 R = SL 2 R/{±I} acts on the upper half-plain H = {z ∈ C | Im(z) > 0} via the Möbius transformation γz = az+b cz+d for γ = a b c d and z ∈ H, so that PSL 2 R = Isom + H holds. Triangle groups are often defined as a subgroup of PSL 2 R generated by reflections on the sides of a triangle in H. However, we here define them as subgroups of SL 2 R to make our argument simple. Put Definition 2.1. The (p, q)-triangle group Γ p,q = Γ(p, q, ∞) is a subgroup of SL 2 R generated by elements S p and U q . This group Γ p,q is a Fuchsian group of the first kind. We especially have Γ 2,3 = SL 2 Z. There is an isomorphism to the amalgamated product Γ p,q ∼ = S p * −I U q ∼ = Z/2pZ * Z/2Z Z/2qZ, which is obtained by applying [Ser03, Theorem 6] to a geodesic segment T = {e πiθ | 1/q θ 1 − 1/p} ⊂ H. We can visualize the group Γ p,q by its fundamental domain in H. Let ∆ = ∆(p, q, ∞) denote the triangle with interior angles π/p, π/q, 0 defined by In addition, let ∆ ′ = ∆ ′ (p, q, ∞) denote the reflection of ∆(p, q, ∞) with respect to the geodesic {e iθ | 0 < θ < π}, that is, we put The vertices a = e πi(1−1/p) , b = e πi/q , and i∞ of ∆(p, q, ∞) are fixed points of S p , U q , and T p,q , respectively. The first two vertices a and b are called elliptic points of Γ p,q , and i∞ is called a cusp of Γ p,q . The stabilizer subgroups of these vertices are given by (Γ p,q ) a = S p , (Γ p,q ) b = U q , and (Γ p,q ) ∞ = ± T p,q , respectively. The two sides of the quadrangle D p,q joined at each elliptic point are Γ p,q -equivalent. Hence, the Riemann surface Γ p,q \H has one cusp, two elliptic points, and genus 0. By the Gauss-Bonnet theorem, or by a direct calculation, one may verify that vol(Γ p,q \H) = Dp,q dxdy y 2 = 2πr pq . In general, an element γ ∈ Γ p,q is said to be elliptic if | tr γ| < 2, parabolic if | tr γ| = 2, and hyperbolic if | tr γ| > 2. In each case, the conjugacy class of γ correspond to elliptic points, the cusp, and closed geodesics on Γ p,q \H respectively (see also Subsection 3.1). An element γ ∈ Γ p,q is said to be primitive if γ = ±σ n for σ ∈ Γ p,q and n ∈ Z implies that n = ±1. We make use of the following lemma in later calculations. Lemma 2.2. For each integer n ∈ Z, let C n (x) ∈ Z[x] denote the Chebyshev polynomial of the second kind characterized by C n (cos t) = sin nt/ sin t, t ∈ R. Then C 0 (x) = 0, C 1 (x) = 1, and C n+1 (x) = 2xC n (x) − C n−1 (x) hold. The generators S p and U q satisfy , U q n = C n+1 (cos π q ) −C n (cos π q ) C n (cos π q ) −C n−1 (cos π q ) . In this subsection, we recollect the notions of meromorphic modular forms and harmonic Maass forms for triangle groups together with some properties. 2.2.1 Meromorphic modular forms For γ = a b c d ∈ SL 2 R and a variable z ∈ H, the automorphic factor is defined by j(γ, z) = cz + d, so that the cocycle condition j(γ 1 γ 2 , z) = j(γ 1 , γ 2 z)j(γ 2 , z) for any γ 1 , γ 2 ∈ SL 2 R holds. For the pair of a function f : H → C and an element γ ∈ SL 2 R, the slash operator of weight k ∈ Z is defined by (f | k γ)(z) = j(γ, z) −k f (γz). Definition 2.3. A meromorphic function f : H → C ∪ {∞} is called a meromorphic modular form of weight k ∈ Z for Γ p,q if the following conditions hold. (i) f | k γ = f for every γ ∈ Γ p,q . (ii) Put λ = 2 cos π p + cos π q and q λ = e 2πiz/λ . Then f (z) has a Fourier expansion of the form n=n ′ a n q n λ , a n ∈ C for some n ′ ∈ Z. If in addition f is holomorphic on H and a n = 0 holds for all n < 0 (resp. n 0), then f is called a holomorphic modular form (resp. cusp form) of weight k for Γ p,q . The space of all holomorphic modular forms of weight k for Γ p,q is denoted by M k (Γ p,q ). Cauchy's residue theorem yields the following valence formula in a similar manner to [Kob93, Proposition 3.8]: Proposition 2.4 (The valence formula). Let f be a non-zero meromorphic modular form of weight k for Γ p,q . Let v P (f ) denote the order of zero of f at each z = P on Γ p,q \H and put v ∞ (f ) = min{n ∈ Z | a n = 0}. Then holds, where a = e πi(1−1/p) and b = e πi/q are the fixed points of S p and U q respectively. We use the following lemma later. Lemma 2.5. M 2 (Γ p,q ) = 0. . Since 0 < pq−p−q pq < 1, we have N = 0, hence p(B + 1) + q(A + 1) − pq = 0. Since p and q are coprime, we may write A + 1 = pl for some l ∈ Z >0 . Now we have 0 = p(B + 1) + pq(l − 1) > 0, hence contradiction. Therefore we have f = 0. The notion of harmonic Maass forms is a generalization of holomorphic modular forms. It was introduced by Bruinier-Funke [BF04] to study geometric theta lifts and played a crucial role in the study of Ramanujan's mock theta functions. It is defined by using ξ-differential operators and the hyperbolic Laplace operators. Definition 2.6. Let k ∈ Z. For a real analytic function f : H → C, the ξ-differential operator ξ k of weight k is defined by where ∂/∂z is Wirtinger's derivative defined by The hyperbolic Laplace operator ∆ k of weight k is defined by A direct calculation yields that holds for any γ ∈ Γ p,q . Hence if f satisfies the modular transformation law f | k γ = f of weight k for every γ ∈ Γ p,q , then so does ξ k f of weight 2 − k. We also note that if f is a holomorphic function, then ξ k f = 0 holds. Definition 2.7. A real analytic function f : H → C is called a harmonic Maass form of weight k ∈ Z for Γ p,q if the following conditions hold. (i) f | k γ = f for every γ ∈ Γ p,q . (ii) ∆ k f (z) = 0. (iii) There exists α > 0 such that f (x + iy) = O(y α ) as y → ∞ uniformly in x ∈ R. The space of all harmonic Maass forms of weight k for Γ p,q is denoted by H k (Γ p,q ). We remark that in a basic textbook of harmonic Maass forms [BFOR17, Definition 4.2], for instance, the condition (iii) is replaced by a slightly different condition, namely, (iii') There exists a polynomial P f (z) ∈ C[q −1 λ ] such that f (z) − P f (z) = O(e −εy ) as y → ∞ for some ε > 0. Whichever condition is chosen, we have M k (Γ p,q ) ⊂ H k (Γ p,q ) and the ξ-differential operator of weight k induces a linear map ξ k : H k (Γ p,q ) → M 2−k (Γ p,q ). A virtue of our choice (iii) is that the function E (p,q), * 2 (z) in Subsection 2.3 will be a harmonic Maass form. Let f ∈ H k (Γ p,q ) and suppose k = 1. Then a standard argument yields a Fourier expansion where c + (n) with n 0 and c − (n) with n 0 are complex constants and W µ,ν (y) denotes the so-called W -Whittaker function (cf. [MOS66, Chapter VII]). If instead k = 1, then y 1−k is replaced by log y. On the other hand, we remark that the Fourier coefficients c − (n) of the remaining nonholomorphic part are closely related to a function ξ k f ∈ M 2−k (Γ p,q ) called the shadow of the mock modular form f + . In fact, we have (p,q), * 2 (z) of weight 2 In this subsection, we construct a harmonic Maass form E (p,q), * 2 (z) and a mock modular form E (p,q) 2 (z) of weight 2 for Γ p,q with explicit descriptions. For this purpose, we first recollect the notion of the Eisenstein series E (p,q) 2k (z, s) of even weight for Γ p,q , that yields most basic examples of harmonic Maass forms. We refer to Iwaniec's book [Iwa02] and Goldstein's paper [Gol73] for some properties, but we rather follow a standard recipe of mock modular forms. Recall that the triangle group Γ p,q < SL 2 R is a Fuchsian group with finite covolume vol(Γ p,q \H) = 2πr/pq and the stabilizer subgroup of the unique cusp i∞ is given by (Γ p,q ) i∞ = ± T p,q . Let λ = 2(cos π p + cos π q ) as before and put σ = λ 1/2 0 0 λ −1/2 ∈ SL 2 R, so that σ is a scaling matrix of the cusp i∞, that is, σi∞ = i∞ and σ −1 T p,q σ = 1 1 0 1 hold. Definition 2.8. Let k be an integer. For z ∈ H and s ∈ C with Re(s) > 1, the real analytic Eisenstein series of weight 2k for Γ p,q is defined by For each s with Re(s) > 1, as a function in z, this series converges absolutely and uniformly on compact subsets of H. By the definition, (E 2k (z, s). The following is classically known. Proposition 2.9 [Iwa02, Proposition 6.13]. The Eisenstein series E (p,q) 0 (z, s) of weight 0 has a meromorphic continuation around s = 1 with a simple pole there with residue The classical Kronecker limit formula describes the constant term of the Eisenstein series E (z, s) in s at s = 1, which is also called the limit function, is given by where C p,q and c p,q (n) are complex numbers described in terms of a certain Dirichlet series. (p,q), * 2 (z) = ξ 0 L p,q (z). Then we have the following. Proof. By a direct calculation, we have (z) satisfies the modular transformation law of weight 2. The conditions (ii) and (iii) in Definition 2.7 are easily verified. Hence we have E (p,q), * 2 (z) ∈ H 2 (Γ p,q ). (z) is a mock modular form of weight 2 and we have The modular transformation law of weight 2 for E (p,q), * 2 (z) yields the modular gap of the function E (p,q) 2 (z) described as follows. . This gap will play a crucial role to define a holomorphic cusp form ∆ p,q (z) in the next subsection. In this subsection, we construct a holomorphic cusp form ∆ p,q (z) of weight 2pq for Γ p,q with no poles and zeros on H. In the course of argument, we introduce a primitive function F p,q (z), a 1-cocycle function R p,q (γ, z), and the Rademacher symbol ψ p,q : Γ p,q → Z as well. where c p,q (n) are those in Proposition 2.10. This F p,q is the regularized primitive function in the sense that the leading coefficient of the Fourier expansion of ∆ p,q (z) = exp F p,q (z) is 1. In addition, let R p,q : Γ p,q × H → C denote the weight 0 modular gap function of F p,q (z) defined by Then we have R p,q (−γ, z) = R p,q (γ, z) and the 1-cocycle relation R p,q (γ 1 γ 2 , z) = R p,q (γ 1 , γ 2 z) + R p,q (γ 2 , z). The Rademacher symbol ψ p,q (γ) By Lemma 2.12, we have d dz (R p,q (γ, z)−2pq log j(γ, z)) = 0. Hence there exists a function ψ p,q : Γ p,q → C satisfying where we assume that Im log j(γ, z) ∈ [−π, π). We call this ψ p,q the Rademacher symbol for Γ p,q . Let us verify that ψ p,q (γ) ∈ Z. Lemma 2.13. For the elements T p,q , S p , U q of Γ p,q (cf. Definition 2.1), we have ψ p,q (T p,q ) = r, ψ p,q (S p ) = −q, and ψ p,q (U q ) = −p. Proof. Let us first show that ψ p,q (U q ) = −p. By the fact that U q q = −I, for any z ∈ H, we have Here the left-hand side is independent of the choice of z, and the righthand side is continuous in z ∈ H. By taking the limit z → i∞ and applying Lemma 2.2, we obtain ψ p,q (U q ) = −p. In a similar way, we may obtain ψ p,q (S p ) = −q. Finally, by we obtain ψ p,q (T p,q ) = r. Proposition 2.14. For any γ ∈ Γ p,q , the value ψ p,q (γ) is an integer. Proof. We prove the assertion by induction on the word length of γ ∈ Γ p,q with respect to the generators S p and U q . We proved in Lemma 2.13 that ψ p,q (S p ), ψ p,q (U q ) ∈ Z. Now suppose that ψ p,q (γ) ∈ Z. If w ∈ {S p , U q }, then we see that 2.4.3 A cusp form of weight 2pq Finally, we define a holomorphic function on H by ∆ p,q (z) = exp F p,q (z). By Proposition 2.14, for any γ ∈ Γ p,q , we have that is, ∆ p,q | 2pq γ = ∆ p,q holds. By the definition, ∆ p,q (z) is holomorphic, and has no zeros and poles on the upper-half plane H. Moreover, by Proposition 2.4, the function vanishes at the cusp i∞. Therefore, by the construction, we have the following. Proposition 2.15. The function ∆ p,q (z) is a cusp form of weight 2pq with a unique zero of order r at the cusp i∞, having a Fourier expansion of the form ∆ p,q (z) = q r λ + O(q r+1 λ ). In addition, Remark 2.16. For the function log η Γp,q,i (z/λ) introduced in [Gol73, Theorem 3.1], we have F p,q (z) = 4pq log η Γp,q,i (z/λ) and ∆ p,q (z) = η Γp,q,i (z/λ) 4pq . However, our ψ p,q (γ) and the generalized Dedekind sum S Γp,q,i (γ) in [Gol73] are slightly different, due to their choices of branches of the logarithm. In terms of our cusp form ∆ p,q (z), the Kronecker limit type formula in Proposition 2.10 is paraphrased as follows. Remark 2.17. The limit function L p,q (z) is an example of polyharmonic Maass forms, that were recently introduced by Lagarias-Rhoades in [LR16] as a generalization of harmonic Maass forms. A real analytic function f : H → C is called a polyharmonic Maass form of weight k ∈ Z and depth r ∈ Z for Γ p,q if it satisfies the conditions (i) and (iii) in Definition 2.7 and (ii)' (∆ k ) r f (z) = 0. In fact, the function L p,q (z) satisfies the above three conditions with k = 0 and r = 2. For further studies on polyharmonic Maass forms, we refer to [Mat19] and [Mat20b] written by the first author. In the previous section, we introduced the Rademacher symbol ψ p,q : Γ p,q → Z by using a certain 1-cocycle function R p,q (γ, z). Let us briefly recall the definition. The harmonic Maass form E (p,q), * 2 (z) yields the mock modular form E (p,q) 2 (z). We defined the regularized primitive function F p,q (z) of 2πirE (p,q) 2 (z) and the cusp form ∆ p,q (z) so that ∆ p,q (z) = exp F p,q (z) = q r λ + O(q r+1 λ ) hold for λ = 2(cos π p + cos π q ). We further put log ∆ p,q (z) = F p,q (z). Our symbol ψ p,q may be defined as follows, assuming that Im log z ∈ [−π, π). Definition 3.1. The Rademacher symbol ψ p,q : Γ p,q → Z is a unique function satisfying Since the classical case ψ 2,3 admits many characterizations as Atiyah and Ghys proved, we may expect that ψ p,q also has many. In this section, we establish characterization theorems of ψ p,q from three aspects; cycle integrals of E (p,q), * 2 (z), a 2-cocycle W generating the bounded cohomology group H 2 b (SL 2 R; R), and an additive character χ p,q : Γ p,q → Z. In addition, we introduce several variants Φ p,q , Ψ p,q , and Ψ h p,q in a view of the classical cases. We obtain several lemmas for our main theorem on the linking number through this section. The group Γ p,q acts on R ∪ {i∞} = ∂H via the Möbius transformation. Let γ ∈ Γ p,q be a hyperbolic element, that is, | tr γ| > 2 holds. Then, there are exactly two fixed points w γ , w ′ γ on c d ∈ Γ p,q is an element with a + d > 2 and c > 0, so that ξ γ > 1 holds. Let S γ denote the geodesic in H connecting two fixed points w γ and w ′ γ . Then the action of γ preserves the set S γ and sends every point on S γ toward w γ . The image S γ of S γ on the Riemann surface (orbifold) Γ p,q \H is a closed geodesic. If in addition γ is primitive, then the arc on S γ connecting any z 0 ∈ S γ and γz 0 is a lift of the simple closed geodesic S γ . Theorem 3.2. Let γ = a b c d ∈ Γ p,q be a primitive element with a + d > 2 and c > 0. Then the cycle integral is given by the Rademacher symbol as Proof. For any z 0 ∈ S γ , the cycle integral coincides with the path integral along S γ on H as We let z 0 = M γ i. Recall that the harmonic Maass form E (p,q), * 2 (z) may be written the sum of holomorphic and non-holomorphic parts as . The integration of the holomorphic part is given by As for the integration of the non-holomorphic part, recall that vol(Γ p,q \H) = 2πr pq . In addition, by changing variables via z = M γ iy, we obtain where we assume that Im log z ∈ [−π, π) as before. By summation, we have which finishes the proof. In this subsection, we give an alternative definition of the Rademacher symbol ψ p,q without use of automorphic forms. We introduce a bounded 2-cocycle W following Asai [Asa70] and prove that ψ p,q is a unique function satisfying 2pqW = −δ 1 ψ p,q . 3.2.1 A 2-cocycle and Asai's sign function Here we introduce a 2-cocycle W corresponding to the universal covering group SL 2 R together with an explicit description with use of Asai's sign function. Definition 3.3. We define a 2-cocycle W : SL 2 R × SL 2 R → Z by W (γ 1 , γ 2 ) = 1 2πi log j(γ 1 , γ 2 z) + log j(γ 2 , z) − log j(γ 1 γ 2 , z) , assuming arg j(γ, z) = Im log j(γ, z) ∈ [−π, π). This is equivalent to say that we have j(γ 1 , γ 2 z)j(γ 2 , z) = j(γ 1 γ 2 , z)e 2πiW (γ 1 ,γ 2 ) in the universal covering group C × of the multiplicative group C × = C − {0}. Since the right-hand side of the definition is continuous in z, the value of W is independent of z. We may easily verify the 2-cocycle condition The universal cover SL 2 R → SL 2 R as manifolds is a group homomorphism as well. The group SL 2 R is called the universal covering group of SL 2 R. Note in addition that each central extension of SL 2 R by Z corresponds to a 2-cocycle SL 2 R × SL 2 R → Z and each isomorphism class of central extensions corresponds to a 2nd cohomology class in H 2 (SL 2 R; Z) (cf. [Bro94, Chapter IV]). Now we have the following. Proposition 3.4. As a group, the universal covering group SL 2 R of SL 2 R is a central extension of SL 2 R by Z corresponding to the 2-cocycle W . In other words, when we identify SL 2 R with SL 2 R × Z as sets, we have (γ 1 , n 1 ) · (γ 2 , n 2 ) = (γ 1 γ 2 , n 1 + n 2 + W (γ 1 , γ 2 )) for every (γ 1 , n 1 ), (γ 2 , n 2 ) ∈ SL 2 R. By virtue of the convention arg j(γ, z) = Im log j(γ, z) ∈ [−π, π), we have W (γ 1 , γ 2 ) ∈ {−1, 0, 1}. Asai introduced the following sign function to explicitly express the values of W . Definition 3.5. For any γ = a b c d ∈ SL 2 R, we define its sign by . The values of W (γ 1 , γ 2 ) are given by the following table. 3.2.2 A 2-coboundary of Γ p,q We next calculate the cohomology of Γ p,q and establish the relation between the 2-cocycle W on Γ p,q and the Rademacher symbol ψ p,q . Lemma 3.8. We have H 1 (Γ p,q ; Z) = H 1 (Γ p,q ; C) = {0} and H 2 (Γ p,q ; Z) ∼ = Z/2pqZ. Proof. Since the triangle group Γ p,q is generated by torsion elements S p and U q , a group homo- The second assertion follows from the facts , and the Mayer-Vietoris sequence for group cohomology. Since H 2 (Γ p,q ; Z) ∼ = Z/2pqZ, we have 2pq[W ] = 0 in H 2 (Γ p,q ; Z). Hence there exists a function f : Γ p,q → Z satisfying the coboundary condition for every γ 1 , γ 2 ∈ Γ p,q . Such f is unique. Indeed, if there are two functions f 1 , f 2 : Γ p,q → C satisfying the same coboundary condition, then the difference f 1 − f 2 is a homomorphism. Hence by H 1 (Γ p,q ; C) = {0}, we have f 1 − f 2 = 0. We further have the following. Theorem 3.9. The Rademacher symbol ψ p,q : Γ p,q → Z is a unique function satisfying Proof. It suffices to verify that the equality ψ p,q (γ 1 γ 2 ) − ψ p,q (γ 1 ) − ψ p,q (γ 2 ) = 2pqW (γ 1 , γ 2 ) holds for every γ 1 , γ 2 ∈ Γ p,q . By the definition of ψ p,q , the left-hand side equals 1 2πi R p,q (γ 1 γ 2 , z) − 2pq log j(γ 1 γ 2 , z) − R p,q (γ 1 , γ 2 z) + 2pq log j(γ 1 , γ 2 z) − R p,q (γ 2 , z) + 2pq log j(γ 2 , z) . The R p,q -terms cancel out by the 1-cocycle relation and the remaining equals 2pqW (γ 1 , γ 2 ). The additive character χ p,q : Γ p,q → Z In this subsection, we provide another characterization of the Rademacher symbol ψ p,q by using an additive character χ p,q : Γ p,q → Z. As before, we assume that SL 2 R = SL 2 R × Z as a set. Let P : SL 2 R → SL 2 R; (γ, n) → γ denote the universal covering map and put Γ p,q = P −1 (Γ p,q ), so that we have Γ p,q = {(γ, n) ∈ SL 2 R | γ ∈ Γ p,q , n ∈ Z}. For each γ ∈ Γ p,q , we define the standard lift by γ = (γ, 0) ∈ Γ p,q . Lemma 3.10. The lifts of S p , U q , T p,q ∈ Γ p,q satisfy S p p = (−I, 1) = U q q , T p,q = −I U q S p . The group Γ p,q is generated by S p and U q . Proof. The equalities immediately follow from the group operation of SL 2 R with use of W and Lemma 2.2. Since we have (I, 1) = S p 2p = U q 2q , the elements S p and U q generate Γ p,q . Let χ : Γ p,q → Z be an additive character, that is, a group homomorphism to the additive group Z. Such χ is determined by the values χ( S p ) = s and χ( U q ) = u. The relation S p p = (−I, 1) = U q q imposes the condition χ(−I, 1) = ps = qu on the pair (s, u). Since p and q are coprime, we have s = mq, u = mp for some m ∈ Z. In addition, since (−I, 1) 2 = (I, 1), we have χ(I, 1) = 2mpq. Define a function V : Γ p,q → Z by putting V (γ) = χ( γ). Then we have χ(γ, n) = χ( γ · (I, 1) n ) = V (γ) + 2mnpq for any (γ, n) ∈ Γ p,q . In addition, for any γ 1 , γ 2 ∈ Γ p,q , by the relation γ 1 · γ 2 = (γ 1 γ 2 , W (γ 1 , γ 2 )), we have If m = −1, then Theorem 3.9 yields V = ψ p,q . Consequently, we obtain the following. Theorem 3.11. The additive character χ p,q : Γ p,q → Z determined by χ p,q ( S p ) = −q and χ p,q ( U q ) = −p satisfies ψ p,q (γ) = χ p,q (γ, n) + 2npq for every γ ∈ Γ p,q and n ∈ Z. Remark 3.12. Theorem 3.11 is a generalization of Asai's result in his unpublished lecture note [Asa03] . His function Φ satisfies Φ(γ) = ψ 2,3 (γ) + 3 sgn(γ) for any γ ∈ SL 2 Z. For the convenience of later use, let us calculate the values of the Rademacher symbol at several elements. By Theorem 3.11, we easily see ψ p,q (−I) = χ pq (−I, 1) + 2pq = −pq + 2pq = pq, ψ p,q (T p,q ) = χ p,q ( −I U q S p ) = pq − p − q = r. The latter agrees with the previous result in Lemma 2.13. In addition, we have the following. Lemma 3.13. For any γ = a b c d ∈ Γ p,q , we have ψ p,q (−γ) = ψ p,q (γ) + pq sgn(γ), Proof. By Theorem 3.9, we have ψ p,q (−γ) = ψ p,q (γ) + ψ p,q (−I) + 2pqW (−I, γ). Recall ψ p,q (−I) = pq. Since W (−I, γ) = 0 if sgn(γ) = +1 and W (−I, γ) = −1 if sgn(γ) = −1, we have ψ p,q (−I) + 2pqW (−I, γ) = pq sgn(γ). In general, the inverse of any (γ, n) ∈ SL 2 R is given by Hence we have ψ p,q (γ −1 ) = χ p,q (γ −1 , 0) = χ p,q ((γ, 1) −1 ) = −χ p,q (γ, 1) = −ψ p,q (γ) + 2pq if c = 0, d < 0, and ψ p,q (γ −1 ) = χ p,q ((γ, 0) −1 ) = −ψ p,q (γ) if otherwise. We also use the following lemma later. Lemma 3.14. Let (x, y) ∈ Z 2 be a pair satifying px + qy = 1, |x| < q, |y| < p, xy < 0 and put In both cases, we have ψ p,q (γ) = 1. Proof. If (p, q) = (2, 3), then we have γ = U −2 3 S 2 = U 3 S −1 2 = 1 1 0 1 . If (p, q) = (2, 3), then by Lemma 2.2, we have By the condition xy < 0, we have c > 0. In addition, we have tr γ = − 2 sin π p · sin π q sin πx q cos π q sin πy p cos π p − sin π q cos πx q sin π p cos πy p + sin πx q sin πy p = 2 sin πx q sin πy p sin π p sin π q cos π p cos π q + 1 + cos πx q cos πy p > 2. For any m, n with 0 < |m| < p and 0 < |n| < q, we have sgn(S p m ) = sgn(m) and sgn(U q n ) = sgn(n). Hence we have ψ p,q (U q −x ) = −xψ p,q (U q ) = px and ψ p,q (S p −y ) = −yψ p,q (S p ) = qy. By xy < 0, we obtain ψ p,q (γ) = ψ p,q (U q −x ) + ψ p,q (S p −y ) + 2pqW (U q −x , S p −y ) = px + qy = 1. In this subsection, we recall several variants of the classical Rademacher symbol and generalize them for any Γ p,q . We modify the Rademacher symbol ψ p,q to obtain a class-invariant function, namely, the original Rademacher symbol Ψ p,q . In addition, we define the Dedekind symbol Φ p,q and the homogeneous Rademacher symbol Ψ h p,q and attach remarks. The classical cases Let us recollect two classical variant Φ 2,3 and Ψ 2,3 of the Rademacher symbol ψ 2,3 . The Dedekind symbol Φ 2,3 : SL 2 Z → Z introduced by Dedekind in 1892 [Ded92] is defined as a unique function satisfying for every γ = a b c d ∈ SL 2 Z and z ∈ H, assuming Im log z ∈ (−π, π). Here, sgn c ∈ {−1, 0, 1} denotes the usual sign function. For each a ∈ Z and c ∈ Z >0 , the Dedekind sum is defined by where we put ((x)) = x − ⌊x⌋ − 1/2 if x ∈ Z and ((x)) = 0 if x ∈ Z. The following formula is due to Dedekind: This symbol Φ 2,3 is not a class-invariant function. In 1956 [Rad56], Rademacher introduced a class-invariant function by modifying the Dedekind symbol, namely, he defined the original Rademacher symbol Ψ 2,3 : SL 2 Z → Z by putting Ψ 2,3 (γ) = Φ 2,3 (γ) − 3 sgn(c(a + d)). This symbol Ψ 2,3 satisfies Ψ 2,3 (γ) = Ψ 2,3 (−γ) = −Ψ 2,3 (γ −1 ) = Ψ 2,3 (g −1 γg) for any γ, g ∈ SL 2 Z. In addition, if tr γ > 0, then log ∆ 2,3 (γz) − log ∆ 2,3 (z) = 12 log j(γ, z) + 2πiΨ 2,3 (γ) holds, that is, we have Ψ 2,3 (γ) = ψ 2,3 (γ). We remark that there are many more variants in literatures with confusions. The clarification between Φ 2,3 and Ψ 2,3 is due to [DIT17]. The original symbol Ψ p,q Let us generalize the original symbol for any Γ p,q . Definition 3.15. We define the original Rademacher symbol Ψ p,q : Γ p,q → Z for Γ p,q by where sgn(γ) ∈ {±1} denotes Asai's sign function and sgn tr γ ∈ {−1, 0, 1} the usual sign function. If we put (p, q) = (2, 3), then we obtain the classical symbol Ψ 2,3 due to Rademacher. If tr γ > 0, then Ψ p,q (γ) = ψ p,q (γ) holds. The following assertion is proved by Lemmas 3.17-3.20. Proposition 3.16. For any γ, g ∈ Γ p,q , Ψ p,q (γ) = Ψ p,q (−γ) = −Ψ p,q (γ −1 ) = Ψ p,q (g −1 γg) holds. In addition, if | tr γ| 2, then Ψ p,q (γ n ) = nΨ p,q (γ) holds for any n ∈ Z. Lemma 3.17. For any γ ∈ Γ p,q , we have Ψ p,q (−γ) = Ψ p,q (γ), that is, Ψ p,q induces a function on Γ p,q /{±I}. Proof. By Lemma 3.13, we obtain Ψ p,q (−γ) = ψ p,q (−γ) + pq 2 sgn(−γ)(1 − sgn tr(−γ)) = ψ p,q (γ) + pq sgn(γ) − pq 2 sgn(γ)(1 + sgn tr γ) = Ψ p,q (γ). Lemma 3.18. For any γ ∈ Γ p,q , we have Ψ p,q (γ −1 ) = −Ψ p,q (γ). Proof. If γ = a b c d ∈ Γ p,q satisfies c = 0 and d < 0, then by Lemma 3.13, Ψ p,q (γ −1 ) = ψ p,q (γ −1 ) − pq = −ψ p,q (γ) + pq = −Ψ p,q (γ). Other cases are obtained in as similar manner. Lemma 3.19. For γ ∈ Γ p,q with | tr γ| 2, we heve Ψ p,q (γ n ) = nΨ p,q (γ). Proof. Since −Ψ p,q (γ −1 ) = Ψ p,q (−γ) = Ψ p,q (γ) holds by the above lemmas, we may assume sgn(γ) > 0, tr γ 2, and n > 0 without loss of generality. Put t = tr γ 2. Then we have γ n = a n (t)γ − a n−1 (t)I, where a 0 (t) = 0, a 1 (t) = 1, and a n (t) = ta n−1 (t) − a n−2 (t). This implies that sgn(γ n ) > 0 and tr(γ n ) > 0 for any n > 0. Hence we obtain Ψ p,q (γ n ) = ψ p,q (γ n ) = χ p,q (γ n , 0) = χ p,q ((γ, 0) n ) = nψ p,q (γ) = nΨ p,q (γ), which conclude the proof. Lemma 3.20. The function Ψ p,q (γ) is a class-invariant function, that is, for any g ∈ Γ p,q , we have Ψ p,q (g −1 γg) = Ψ p,q (γ). Proof. We may assume sgn(γ) > 0 and tr γ 0 without loss of generality. It suffices to show the equation Ψ p,q (g −1 γg) = Ψ p,q (γ) for generators g = T p,q , S p . By the definitions, we have Ψ p,q (g −1 γg) = Ψ p,q (g −1 ) + Ψ p,q (γ) + Ψ p,q (g) + 2pq(W (g −1 , γg) + W (γ, g)) + pq 2 sgn(g −1 γg)(1 − sgn tr(g −1 γg)) − sgn(g −1 )(1 − sgn tr(g −1 )) − 1 + sgn tr γ − sgn(g)(1 − sgn tr g) . By Ψ p,q (g −1 ) + Ψ p,q (g) = 0 and we obtain Ψ p,q (g −1 γg) =Ψ p,q (γ) + pq 2 − sgn(g −1 ) sgn(γg) sgn(g −1 γg) − sgn(g) sgn(γg) − sgn(g −1 γg) sgn tr γ + sgn(g −1 ) sgn tr(g −1 ) + sgn tr γ + sgn(g) sgn tr g . If g = T p,q , then we have sgn(γg) = sgn(g −1 γg) = sgn(γ) = 1, that is, Ψ p,q (g −1 γg) = Ψ p,q (γ). If g = S p , then we have Ψ p,q (g −1 γg) =Ψ p,q (γ) + pq 2 sgn(γg) − sgn tr γ sgn(g −1 γg) − 1 . Assume γ = a b c d with a + d 0 and sgn(γ) > 0. Then we see that (i) If a+d = 0, then −bc = a 2 +1 > 0. Thus we have c > 0 and −b > 0, that is, sgn(g −1 γg) = 1. (ii) If a + d > 0, then it suffices to show that (sgn(γS p ) − 1)(sgn(S −1 p γS p ) − 1) = 0. -If d > 0, then we have sgn(γS p ) = 1. -If d = 0, then sgn(γS p ) = sgn(−c) < 0. In addition, by det(γ) = −bc = 1, we have b < 0. Hence we obtain sgn(S p −1 γS p ) = 1. -If d < 0, then sgn(γS p ) = −1. In this case, we have a > 0, c > 0, and b < 0. Hence we have sgn(S p −1 γS p ) = 1. In conclusion, we obtain Ψ p,q (g −1 γg) = Ψ p,q (γ) for all cases. Other variants Φ p,q and Ψ h p,q Here, we discuss two more variants Φ p,q and Ψ h p,q . Definition 3.21. We define the Dedekind symbol Φ p,q : Γ p,q → 1 2 Z by Φ p,q (γ) = Ψ p,q (γ) + pq 2 sgn(c(a + d)). This symbol Φ p,q is a unique function satisfying for every γ i = * * c i * ∈ Γ p,q with γ 1 γ 2 = * * c 12 * , hence a generalization of [RG72, (62)] . The values at generators are given by Definition 3.22. We define the homogeneous Rademacher symbol Ψ h p,q : Γ p,q → Z by the homogenization of ψ p,q , that is, we put In comparison with Proposition 3.16, for any γ, g ∈ Γ p,q and n ∈ Z, we have . If | tr γ| 2, then Ψ h p,q (γ) = Ψ p,q (γ) holds. If instead | tr γ| < 2, then we have Ψ h p,q (γ) = 0, while the original symbol satisfies Note that we have tr S p = 2 cos π p and tr U q = 2 cos π q . If tr γ 2, then Ψ h p,q (γ) = ψ p,q (γ) holds. Remark 3.23. Recently, in a view of the Manin-Drinfeld theorem, Burrin [Bur20] introduced certain functions for a general Fuchsian group Γ by using a recipe close to ours. Her functions may be seen as generalizations of our Φ p,q and Ψ p,q , for which our Theorem 3.2 persist. She also proved that if Γ is a non-cocompact Fuchsian group with genus zero, then the values of the functions are in Q. Our result further claims for Γ p,q that the values are in Z. In this section, we establish our main result, that is, the coincidence of the values of the Rademacher symbol and the linking number between modular knots and the (p, q)-torus knot. Here, we prepare group theoretic lemmas, which enable us to clearly recognize the natural Z/rZcover h : S 3 − K p,q ։ L(r, p − 1) − K p,q , as well as to make an explicit argument. Recall that the universal covering group SL 2 R is the central extension of SL 2 R by Z corresponding to the 2-cocycle W , that is, SL 2 R is SL 2 R × Z as a set and endowed with the multiplication (γ 1 , n 1 ) · (γ 2 , n 2 ) = (γ 1 γ 2 , n 1 + n 2 + W (γ 1 , γ 2 )). Let P : SL 2 R → SL 2 R; (γ, n) → γ denote the natural projection and put Γ p,q = P −1 (Γ p,q ), so that we have Γ p,q = {(γ, n) | γ ∈ Γ p,q } < SL 2 R. For each γ ∈ SL 2 R, define the standard lift by γ = (γ, 0) ∈ SL 2 R. Then Γ p,q is generated by S p and U q , for which S p p = U q q = (−I, 1) holds. Recall r = pq − p − q. We here explicitly define a discrete subgroup G r < SL 2 R by The following lemmas are due to Tsanov [Tsa13] . Since the original assertions are for PSL 2 R, we partially attach proofs for later use. For each group G, let Z(G) denote the center, [G, G] the commutator subgroup, and G ab the abelianization. (2) Since r is an odd number coprime to both p and q, there exist some s, t ∈ Z satisfying rs ≡ 1 (mod 2p) and rt ≡ 1 (mod 2q), hence S p rs = S p and U q rt = U q . Thus we have S p , U q ∈ P (G r ). (2) Γ p,q ab ∼ = G r ab ∼ = Z. (3) Γ p,q /G r ∼ = Γ p,q ab /G r ab ∼ = Z/rZ. As mentioned in Section 1, we have the following. (1) The spaces Γ p,q \ SL 2 R ∼ = Γ p,q \ SL 2 R are homeomorphic to the exterior of a knot K p,q in the lens space L(r, p−1), where K p,q is the image of a (p, q)-torus knot via the Z/rZ-cover S 3 ։ L(r, p − 1). (2) The space G r \ SL 2 R is homeomorphic to the exterior of the torus knot K p,q in S 3 . The second assertion was established by Raymond-Vasquez by using the theory of Seifert fibrations in [RV81]. Tsanov gave explicit homeomorphisms for both cases in [Tsa13] . We remark that Tsanov discussed the lens space L(r, p(q 1 −p 1 +pp 1 )) for a pair (p 1 , q 1 ) ∈ Z 2 with pp 1 +qq 1 = 1, which is homeomorphic to L(r, p − 1) by Brody's theorem. Since the fundamental group is given by π 1 (Γ p,q \ SL 2 R) ∼ = π 1 ( Γ p,q \ SL 2 R) ∼ = Γ p,q , by the Hurewicz theorem and the lemmas above, we obtain the following. Lemma 4.4. The groups G r ∼ = π 1 (S 3 − K p,q ) are the kernels of any surjective homomorphism Γ p,q ∼ = π 1 (L(r, p − 1) − K p,q ) ։ Z/rZ. We may identify the corresponding Z/rZ-cover h : S 3 − K p,q → L(r, p − 1) − K p,q with the natural surjection G r \ SL 2 R ։ Γ p,q \ SL 2 R. The groups G r ab ∼ = H 1 (S 3 −K p,q ; Z) ∼ = Z may be seen as the subgroups of Γ p,q ab ∼ = H 1 (L(r, p− 1) − K p,q ; Z) ∼ = 1 r Z of index r in a natural way. The following diagram visualizes the situation. Here, for G = Γ p,q and G r , G ′ denotes the commutator subgroup of G and Z ′ (G) denotes the subgroup of Z(G) ∼ = Z with index 2. The Z-covers of L(r, p − 1) − K p,q and S 3 − K p,q are denoted by L ∞ = X ∞ . In this subsection, we introduce the notion of modular knots for Γ p,q around the (p, q)-torus knot in the lens space L(r, p − 1), recall the notions of the linking number and the winding number, and establish the former half of our main result on the linking number. Definition 4.5. (1) Let γ = a b c d ∈ Γ p,q be a primitive element with a + d > 2 and c > 0, so that γ is diagonalized by the scaling matrix M γ and its larger eigenvalue satisfies ξ γ > 1. Define an oriented simple closed curve C γ (t) in Γ p,q \ SL 2 R by We call the image C γ in Γ p,q \ SL 2 R ∼ = L(r, p − 1)− K p,q with the induced orientation the modular knot associated to γ. (2) Let γ ∈ Γ p,q be any hyperbolic element, so that we have γ = ±γ n 0 for some primitive element γ 0 = a b c d ∈ Γ p,q with a + d > 2 and c > 0, and n ∈ Z. We define the modular knot associated to γ by C γ = nC γ 0 with multiplicity. Linking numbers A general theory of the linking number in a rational homology 3sphere can be found in [ST80, Section 77]. Since H 1 (L(r, p − 1); Z) ∼ = Z/rZ, the linking number in L(r, p−1) takes value in 1 r Z. Via a standard homeomorphism Γ p,q \ SL 2 R ∼ = → T 1 (Γ p,q \H) to the unit tangent bundle, the knot K p,q may be seen as the cusp orbit with a natural orientation. Let µ be a standard meridian of K p,q and consider the isomorphism H 1 (L(r, p − 1) − K p,q ; Z) ∼ = → 1 r Z sending [µ] to 1. A standard meridian µ may be explicitly given by the curve c(t) in the proof of Proposition 4.9 with 0 t λ. Definition 4.6. The linking number lk(K, K p,q ) of an oriented knot K in L(r, p − 1) − K p,q and the knot K p,q is defined as the image of [K] via the isomorphism H 1 (L(r, p − 1) − K p,q ; Z) ∼ = → 1 r Z. This definition naturally extends to a knot with multiplicity, that is, a formal sum of knots with coefficients in Z. In order to compute the linking number, let us recall the notion of the winding number. Let the unit circle T = {|z| = 1} ⊂ C be endowed with the counterclockwise orientation and let H 1 (C × ; Z) ∼ = → Z denote the isomorphism sending [T] to 1. Definition 4.7. For an oriented closed curve C in C × , the winding number ind(C, 0) ∈ Z is defined to be the image of [C] via the isomorphism H 1 (C × ; Z) ∼ = → Z. Equivalently, it is defined by the cycle integral as The equivalence of two definitions is verified by Cauchy's integral theorem. We define a lift ∆ p,q : SL 2 R → C × of the cusp form ∆ p,q (z) by ∆ p,q (g) = j(g, i) −2pq ∆ p,q (gi). Since ∆ p,q (z) has no zeros on H and satisfies ∆ p,q (γg) = ∆ p,q (g) for any γ ∈ Γ p,q , we obtain the induced continuous function ∆ p,q : Γ p,q \ SL 2 R → C × . Proposition 4.8. For a modular knot C γ defined in Definition 4.5 (1), we have ind( ∆ p,q (C γ ), 0) = ψ p,q (γ). Proof. Recall d dz log ∆ p,q (z) = 2πirE (p,q) 2 (z) and put z 0 = M γ i. Then by Theorem 3.2, we obtain Proposition 4.9. The function ∆ p,q induces an isomorphism H 1 (Γ p,q \ SL 2 R; Z) Proof. The function ∆ p,q induces a group homomorphism ( ∆ p,q ) * : H 1 (Γ p,q \ SL 2 R; Z) → H 1 (C × ; Z). Since both homology groups are isomorphic to Z, it suffices to show the surjectivity. If (p, q) = (2, 3), take a sufficiently large y ∈ R >0 . Define a closed curve in SL 2 Z\ SL 2 R by C y (t) = 1 t 0 1 and that in C × by ∆ 2,3 (C y (t)) = y 6 ∆ 2,3 (t + iy), (0 t 1). Since ∆ 2,3 (z) = q 1 + O(q 2 1 ), we have ind( ∆ 2,3 (C y (t)), 0) = 1. Thus the map ( ∆ 2,3 ) * is surjective. If (p, q) = (2, 3), take the hyperbolic element γ ∈ Γ p,q defined in Lemma 3.14. By Proposition 4.8, we have ind( ∆ p,q (C γ ), 0) = ψ p,q (γ) = 1, which concludes that ( ∆ p,q ) * is surjective. Theorem in L(r, p − 1) By Proposition 4.9, for any oriented knot K in L(r, p − 1) − K p,q ∼ = Γ p,q \ SL 2 R, we have lk(K, K p,q ) = 1 r ind( ∆ p,q (K), 0). Together with the results in Subsection 3.4, we conclude the following. Theorem 4.10. (1) Let γ = a b c d ∈ Γ p,q be a primitive element with a + d > 2 and c > 0. Then the linking number of the modular knot C γ and the image K p,q of the (p, q)-torus knot in the lens space L(r, p − 1) is given by lk(C γ , K p,q ) = 1 r ψ p,q (γ). (2) Let γ ∈ Γ p,q be any hyperbolic element. Then the linking number is given by In this subsection, we investigate modular knots around the (p, q)-torus knot K p,q in S 3 to establish the latter half of our main theorem on the linking number. Definition 4.11. For an oriented knot K in S 3 − K p,q , the linking number lk(K, K p,q ) ∈ Z is defined by the image of [K] via the isomorphism H 1 (S 3 − K p,q ; Z) ∼ = → Z sending a standard meridian µ of K p,q to 1. This definition naturally extends to knots with multiplicity. Recall that the restriction of the Z/rZ-cover h : S 3 ։ L(r, p − 1) to the exterior of K p,q may be identified with the natural surjection G r \ SL 2 R ։ Γ p,q \ SL 2 R. Let K be an oriented knot in L(r, p − 1) − K p,q and K ′ a connected component of h −1 (K). The following two lemmas are consequences of a standard argument of the covering theory (e.g., the lifting property of continuous maps, [Hat02, Propositions 1.33, 1.34]). Lemma 4.12. The covering degree of the restriction h : K ′ → K coincides with the order of [K] in H 1 (L(r, p − 1); Z) ∼ = Z/rZ. The covering degree of h : K p,q → K p,q is equal to r. Proof. Note that the decomposition group of K ′ is a subgroup of the Deck transformation group Deck(h) ∼ = H 1 (L(r, p − 1); Z) ∼ = Z/rZ generated by [K] . The assertion follows from the Hilbert ramification theory for Z/rZ-cover [Uek14, Section 2]. Lemma 4.13. If [K] in H 1 (L(r, p − 1); Z) ∼ = Z/rZ is of order m, then we have lk(K ′ , K p,q ) = m lk(K, K p,q ). Proof. We have a connected surface Σ in L(r, p − 1) with ∂Σ = mK and a connected component Σ ′ of the preimage h −1 (Σ) with ∂Σ ′ = K ′ . Let ι denote the intersection number. Then by Lemma 4.12, we have lk(K ′ , K p,q ) = ι(Σ ′ , K p,q ) = ι(Σ, K p,q ) = lk(mK, K p,q ) = m lk(K, K p,q ). 4.3.2 Modular knots in Z/rZ-cover We define a modular knot in S 3 as a connected component of the inverse image of that in L(r, p − 1). Definition 4.14. (1) Let γ = a b c d ∈ Γ p,q be a primitive element with a + d > 2 and c > 0. Consider the modular knot C γ in L(r, p − 1) − K p,q associated to γ and let m γ denote the order of [C γ ] in H 1 (L(r, p − 1); Z) ∼ = Z/rZ, so that the inverse image h −1 (C γ ) consists of exactly r/m γ -connected components. We call each connected component C ′ γ of h −1 (C γ ) a modular knot associated to γ ∈ Γ p,q in S 3 − K p,q . (2) Let γ ∈ Γ p,q be any hyperbolic element, so that we have γ = ±γ ν 0 for some primitive γ 0 = a b c d ∈ Γ p,q with a + d > 2 and c > 0 and ν ∈ Z. Let C ′ γ 0 be a modular knot in S 3 − K p,q associated to γ 0 . We call the knot C ′ γ = νC ′ γ 0 with multiplicity a modular knot associated to γ ∈ Γ p,q in S 3 − K p,q . The following lemma plays a key role to explicitly find the integer m γ . Lemma 4.15. For each γ ∈ Γ p,q , we have (γ, n) ∈ G r if and only if 2pqn ≡ ψ p,q (γ) mod r holds. Such n's define an element in Z/rZ. If n γ ∈ Z with (γ, n γ ) ∈ G r , then gcd(r, n γ ) = gcd(r, ψ p,q (γ)) holds. Proof. By Lemma 4.1 (2), there exitst some n ∈ Z satisfying (γ, n) ∈ G r . In addition, by Lemma 4.1 (1), we have Z ′ (G r ) = P −1 (I) ∩ G r = (I, r) = (−I, 1) 2r , which is the subgroup of Z(G r ) ∼ = Z with index 2. Now suppose (γ, n), (γ, n ′ ) ∈ G r . Then we have (γ, n)(γ, n ′ ) −1 ∈ G r , which implies n − n ′ ≡ 0 mod r. Thus the set of n ∈ Z with (γ, n) ∈ G r defines a class n γ ∈ Z/rZ. Now take n γ ∈ Z with (γ, n γ ) ∈ G r for each γ ∈ Γ p,q , so that we have a map n • : Γ p,q → Z. Note that gcd(2pq, r) = 1. Since Γ p,q is generated by S p and U q of orders 2p and 2q, a group homomorphism Γ p,q → Z/rZ is trivial, that is, we have H 1 (Γ p,q ; Z/rZ) = 0. Since (γ 1 , n γ 1 ) · (γ 2 , n γ 2 ) = (γ 1 γ 2 , n γ 1 + n γ 2 + W (γ 1 , γ 2 )) in G r , we have n γ 1 γ 2 ≡ n γ 1 + n γ 2 + W (γ 1 , γ 2 ) mod r. On the other hand, by Theorem 3.9, we have ψ p,q (γ 1 γ 2 ) = ψ p,q (γ 1 ) + ψ p,q (γ 2 ) + 2pqW (γ 1 , γ 2 ). Hence we have a group homomorphism ψ p,q (γ) − 2pqn γ mod r : Γ p,q → Z/rZ, which must be zero by H 1 (Γ p,q ; Z/rZ) = 0. Thus we obtain 2pqn γ ≡ ψ p,q (γ) mod r. Again by gcd(2pq, r) = 1, we obtain gcd(r, n γ ) = gcd(r, ψ p,q (γ)). Now let γ = a b c d ∈ Γ p,q be a primitive element with a + d > 2 and c > 0 and take n γ ∈ Z with (γ, n γ ) ∈ G r . Lemma 4.16. For each l ∈ Z/rZ, we may define a simple closed curve in G r \ SL 2 R by C γ,l (t) = M γ e t 0 0 e −t , l , (0 t r gcd(r, ψ p,q (γ)) log ξ γ ). Proof. Note that we have Then a direct calculation yields Hence for any k ∈ Z, we have C γ,l (t + k log ξ γ ) = (I, −kn γ )C γ,l (t). Since Z ′ (G r ) = P −1 (I) ∩ G r = (I, r) , we have (I, −kn γ ) ∈ G r if and only if −kn γ = 0 in Z/rZ holds. The least positive k with −kn γ = 0 is given by k = r/gcd(r, n γ ) = r/gcd(r, ψ p,q (γ)). Hence we obtain the assertion. The image C γ,l in G r \ SL 2 R ∼ = S 3 − K p,q with the induced orientation is a modular knot associated to γ. Proposition 4.17. For l, l ′ ∈ Z/rZ, we have C γ,l = C γ,l ′ if and only if l ≡ l ′ mod gcd(r, ψ p,q (γ)) holds. The set of modular knots in S 3 − K p,q associated to γ coincides with {C γ,l | l ∈ Z/rZ} = {C γ,l | l = 0, 1, · · · , gcd(r, ψ p,q (γ)) − 1}. Proof. Suppose C γ,l = C γ,l ′ . Then there exists some t ∈ R >0 satisfying C γ,l (0) = C γ,l ′ (t) in G r \ SL 2 R, that is, there exists some (σ, s) ∈ G r satisfying (σ, s)(M γ , l) = M γ e t 0 0 e −t , l ′ . Since σM γ = M γ e t 0 0 e −t , there exists some k ∈ Z >0 satisfying σ = γ k , t = k log ξ γ , and s ≡ kn γ mod r. Since we have kn γ + l ≡ l ′ mod r. Hence we have l ≡ l ′ mod gcd(r, n γ ) = gcd(r, ψ p,q (γ)). Suppose instead that l ≡ l ′ mod gcd(r, n γ ). Then we have l ′ = l+k gcd(r, n γ ) and gcd(r, n γ ) = ar + bn γ for some k, a, b ∈ Z. By we obtain C γ,l = C γ,l ′ . Comparing the covering degree, we obtain the second assertion. Proposition 4.18. The element [C γ ] ∈ H 1 (L(r, p − 1); Z) ∼ = Z/rZ is of order m γ = r gcd(r, n γ ) = r gcd(r, ψ p,q (γ)) . Proof. Since the period of C γ (t) is log ξ γ , Lemma 4.16 yields that the covering degree of the restriction h : C γ,l → C γ is r/ gcd(r, ψ p,q (γ)). By Lemma 4.12, we obtain the assertion. 4.3.3 Theorem in S 3 By Lemma 4.13, Theorem 4.10, and by Proposition 4.18, we obtain lk(C ′ γ , K p,q ) = m γ lk(C γ , K p,q ) = m γ r ψ p,q (γ) = 1 gcd(r, ψ p,q (γ)) ψ p,q (γ). Together with the results in Subsection 3.4, we conclude the following. the winding number of ∆ p,q (f a (t)) (0 t π p ) around the origin is −q. In a similar way, the winding number of ∆ p,q (f b (t)) (0 t π q ) is −p. Thus by Lemma 2.13, we see that and Theorem 4.10 (1) for the Rademacher symbol ψ p,q may (literally) extends to these curves. On the other hand, for any non-elliptic point z = x + iy ∈ H, the corresponding fiber (a generic fiber) in L(r, p − 1) − K p,q ∼ = Γ p,q \ SL 2 R is parametrized as Indeed, we have f z (π) = −f z (0) = f z (0) and f z (t)i = z. By the winding number of ∆ p,q (f z (t)) (0 t π) around the origin is ind( ∆ p,q (f z ), 0) = −pq. Hence the linking number of a generic fiber is given by lk(f z , K p,q ) = −pq r . Knots for S p , U q , and T p,q In order to extend the the theorems on linking numbers to whole Γ p,q , we define knots corresponding to elliptic and parabolic elements. Take a sufficiently small ε ∈ R >0 . For the elliptic point a = e πi(1−1/p) , we consider a circlẽ with a clockwise orientation, where d hyp denotes the hyperbolic distance on H. The elliptic element S p acts onc a as a rotation of angle −2π/p. Take any point z 0 ∈c a and let s a denotes the circle segment connecting z 0 to S p z 0 . Then the image c a of s a in Γ p,q \H is a simple closed curve. In addition, take any point Z 0 ∈ SL 2 R with Z 0 i = z 0 and let s a denote the section of s a connecting Z 0 to S p Z 0 . Then the image C a of s a in Γ p,q \ SL 2 R ∼ = L(r, p − 1) − K p,q is a simple closed curve satisfying C a i = c a . Since C a → f a as ε → 0, we have lk(C a , K p,q ) = lk(f a , K p,q ) = −q r . Similarly, for b = e πi/q , we define simple closed curves c b and C b satisfying C b i = c b and For the parabolic element T p,q , as in the proof of Proposition 4.9 for (p, q) = (2, 3), we take a lift C y (t) (0 t λ = 2(cos π p + cos π q )) of a holocycle so that we have lk(C y , K p,q ) = 1 = r r = 1 r ψ p,q (T p,q ). Theorem on whole Γ p,q Note that the fundamental group of the orbifold Γ p,q \H is described by both the languages of loops and covering spaces (cf.[Rat19, Chapter 13]). For each γ ∈ Γ p,q , let w be a fixed point on H ∪ R ∪ {i∞} and consider the stabilizer (Γ p,q ) w . If γ is hyperbolic or parabolic, then (Γ p,q ) w ∼ = Z × Z/2Z. If instead γ is elliptic, then (Γ p,q ) w is a finite cyclic group. Letc be a curve in H which is stable under the action of (Γ p,q ) w and let c denote the image ofc in Γ p,q \H. If γ is elliptic, then c is a cycle around a cone point. If γ is parabolic, Remark 4.26. Ghys claims in [Ghy07, Section 3.4] that if we adapt the definition of modular knots to parabolic and elliptic elements, then his theorem follows from results of Atiyah [Ati87] and , which explicitly investigate Euler cocycles in a view of Homeo + S 1 . If we directly extend the results of Atiyah and Barge-Ghys for Γ p,q , then we may obtain alternative proofs of our theorems on the linking numbers. Finally, we give some remarks and further problems. Ghys gave three proofs for his theorem on the Rademacher symbol for SL 2 Z and the linking number around the trefoil. In this article, through Sections 2-4, we generalized his first proof in [Ghy07, Section 3.3] by introducing the cusp form ∆ p,q (z), as well as discussed an Euler cocycle in a view of his second outlined proof in [Ghy07, Section 3.4]. Ghys's third proof in [Ghy07, Section 3.5] is a dynamical approach. A Lorenz knot is a periodic orbit appearing in the Lorenz attractor. Ghys proved for SL 2 Z that isotopy classes of Lorenz knots and modular knots coincide. In addition, he gave an explicit formula for lk(C γ , K 2,3 ) by using the Lorenz template. A hyperbolic element γ ∈ SL 2 Z is conjugate to a matrix of the form γ ∼ ±S 2 U ε 1 3 S 2 U ε 2 3 · · · S 2 U εn 3 with ε i ∈ {+1, −1}. Then, the linking number counts the number of left and right codes on the Lorenz template, that is, On the other hand, Rademacher showed in [RG72, (70)] that n i=1 ε i = Ψ 2,3 (γ). Thus we obtain lk(C γ , K 2,3 ) = Ψ 2,3 (γ). The templates for geodesic flows for triangle groups are studied by Dehornoy and Pinsky [Pin14, Deh15, DP18] . In particular, Dehornoy [Deh15, Proposition 5.7] gave an explicit formula for the linking number between a periodic orbit of the geodesic flow Φ Γp,q\H and the (p, q)-torus knot K p,q . By combining their result and Theorem 4.10, we may obtain an explicit formula of the Rademacher symbol Ψ p,q (γ). On the other hand, if one can show the explicit formula of Ψ p,q (γ) directly from the definition, then we obtain a generalization of Ghys's third proof. It is a natural question to ask the relation between the linking number lk(C γ , K p,q ) of a modular knot and the length ℓ(C γ ) of the corresponding closed geodesic on the modular orbifold. Based on Sarnak's idea in his letter [Sar10] , Mozzochi [Moz13] proved variants of prime geodesic theorems to establish the following distribution formula, invoking the Selberg trace formula for SL 2 Z; Proposition 5.1. Suppose that γ runs through conjugacy classes of primitive hyperbolic elements in SL 2 Z with tr γ > 2 and let ℓ(γ) = 2 log ξ γ denote the length of the image of each modular knot C γ in SL 2 Z\H. Then for each −∞ a b ∞, we have lim y→∞ #{γ | ℓ(C γ ) y, a lk(C γ , K 2,3 ) ℓ(C γ ) b} #{γ | ℓ(C γ ) y} = arctan πb 3 − arctan πa 3 π . Von Essen generalized their results in his PhD thesis [vE14] for any cofinite Fuchsian group with a multiplier system; Let Γ < SL 2 R be a cofinite Fuchsian group, let f : H → C be a holomorphic modular form of weight 1 for Γ with no zero on H, and let ν : Γ → C be a multiplier system, namely, we have f (γz) = ν(γ)j(γ, z)f (z) for every γ ∈ Γ. For its harmonic logarithm F (z) = log f (z), define Φ : Γ → C by F (γz) − F (z) = log j(γ, z) + 2πiΦ(γ). Assume in addition that the image of Φ is contained in Q. By invoking the Selberg trace formula for Fuchsian groups, von Essen gave generalizations of Sarnak-Mozzochi's results. For instance, his Theorem H implies the following. Proposition 5.2. If we replace SL 2 Z by Γ in Proposition 5.1, then we have lim y→∞ #{γ | ℓ(γ) y, a Φ(γ) ℓ(γ) b} #{γ | ℓ(γ) y} = arctan 4πb − arctan 4πa π . We remark that von Essen also showed for the Hecke triangle group H n = Γ 2,n a formula which is essentially the same as in our Theorem 4.10 (1). His construction of the cusp form ∆ 2,n (z) differs from ours but is closely related to Tsanov's construction of ω ∞ (z, dz) explained in Remark 4.20. His results and Proposition 5.2 are applicable to our setting with a more general triangle group Γ p,q . In fact, let f (z) = ∆ p,q (z) 1/2pq = exp 1 2pq F p,q (z) and F (z) = 1 2pq F p,q (z). By Definition 3.1, we have f (γz) = ν(γ)j(γ, z)f (z), ν(γ) = e 2πi ψp,q (γ) 2pq , and Φ(γ) = 1 2pq ψ p,q (γ). Thus, we obtain the following. Corollary 5.3. If we replace SL 2 Z by Γ p,q in Proposition 5.1, then we have lim y→∞ #{γ | ℓ(γ) y, a ψ p,q (γ) ℓ(γ) b} #{γ | ℓ(γ) y} = arctan 2πb pq − arctan 2πa pq π . By our Theorem 4.10, we may replace ψ p,q (γ) by rlk(C γ , K p,q ) to obtain the Sarnak-Mozzochi formula for Γ p,q . Remark 5.4. The set of modular knots around the trefoil satisfies another distribution formula called the Chebotarev law in the sense of Mazur [Maz12] and McMullen [McM13] , so that it may be seen as an analogue of the set of all prime numbers in SpecZ [Uek21a, Uek21b] , in a sense of arithmetic topology [Mor12] . An exploration of a unified viewpoint for these formulas would be of further interest. 5.3.1 Hyperbolic analogue Duke-Imamoḡlu-Tóth [DIT17] investigated the linking number of two modular knots for SL 2 Z. More precisely, they introduced a hyperbolic analogue of the Rademacher symbol Ψ γ (σ) for two hyperbolic elements γ, σ ∈ SL 2 Z by using rational period functions, and established the equation Ψ γ (σ) = lk(C + γ + C − γ , C + σ + C − σ ). Here C + γ is the modular knot as before, and C − γ is another knot such that C + γ + C − γ is null-homologous in S 3 − K 2,3 . Furthermore, the first author [Mat20a] gave an explicit formula for the hyperbolic Rademacher symbol Ψ γ (σ) in terms of the coefficients of the continued fraction expansion of the fixed points of γ and σ. An open question for SL 2 Z is to find a modular object yielding the linking number lk(C γ , C σ ) (see also [Ric21] ). We may expect similar results for general triangle groups Γ(p, q, r). In [Ati87, Theorem 5 .60], Atiyah gave seven different definitions of the Rademacher symbol for hyperbolic elements of SL 2 Z (see also [BG92] ). It would be interesting to extend any of them for Γ p,q . Since torus knots are algebraic knots, we have a natural action of the absolute Galois group on the profinite completions of the knot groups. We wonder if we may, in a sense, parametrize the Galois action via modular knots. The reciprocity of Dedekind sums and the factor set for the universal covering group of SL(2, R) Rademacher's Φ-function The logarithm of the Dedekind η-function On two geometric theta lifts Harmonic Maass forms and mock modular forms: theory and applications Cohomology of groups The Manin-Drinfeld theorem and the rationality of Rademacher symbol Erläuterungen zu zwei fragmenten von. Riemann, B. Riemanns gesammelte math. Werke und wissenschaftlicher Nachlaß. 2. Auflage (1892) Geodesic flow, left-handedness and templates Automorphic forms for triangle groups Modular cocycles and linking numbers Coding of geodesics and Lorenz-like templates for some geodesic flows Bounded cohomology of discrete groups Ghy07Étienne Ghys, Knots and dynamics, International Congress of Mathematicians Dedekind sums for a Fuchsian group. I, Nagoya Math Algebraic topology Spectral methods of automorphic forms Introduction to elliptic curves and modular forms Polyharmonic Maass forms for PSL Traces of CM values and cycle integrals of polyharmonic Maass forms Mat20b , Polyharmonic weak Maass forms of higher depth for SL 2 (Z) Lecture notes for the conference "Geometry, Topology and Group Theory Knots which behave like the prime numbers MR 0418127 MM85 Shigenori Matsumoto and Shigeyuki Morita, Bounded cohomology of certain groups of homeomorphisms An introduction to arithmetic topology Formulas and theorems for the special functions of mathematical physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften Linking numbers of modular geodesics Templates for geodesic flows, Ergodic Theory Dynam Foundations of hyperbolic manifolds The Mathematical Association of America MR 4259279 RV81 Frank Raymond and Alphonse T. Vasquez, 3-manifolds whose universal coverings are Lie groups Linking numbers of modular knots Topology of 3-dimensional fibered spaces Triangle groups, automorphic forms, and torus knots On the homology of branched coverings of 3-manifolds Chebotarev links are stably generic Automorphic forms -multiplier systems and Taylor coefficients Eine arithmetische Eigenschaft automorpher Formen zu gewissen nichtarithmetischen Gruppen 744 Motooka, Nishi-ku, Fukuoka-shi The authors would like to express their sincere gratitude to Masanobu Kaneko for his introduction to Asai's work in a private seminar and to Masanori Morishita for posing an interesting question related to Ghys's work. The authors are also grateful to Pierre Dehornoy, Kazuhiro Ichihara,Özlem Imamoḡlu, Morimichi Kawasaki, Ulf Kühn, Shuhei Maruyama, Makoto Sakuma, Yuji Terashima, and Masahito Yamazaki for useful information and fruitful conversations. Furthermore, the authors would like to thank all the participants who joined the online seminar FTTZS throughout the COVID-19 situation for cheerful communication. The first and the second authors have been partially supported by JSPS KAKENHI Grant Number JP20K14292 and JP19K14538 respectively. The Rademacher symbols for triangle groups Theorem 4.19. (1) Let γ = a b c d ∈ Γ p,q be a primitive hyperbolic element with tr γ > 2 and c > 0. Then the linking number of each modular knot C ′ γ in S 3 − K p,q associated to γ and the (p, q)-torus knot K p,q is given by lk(C ′ γ , K p,q ) = 1 gcd(r, ψ p,q (γ)) ψ p,q (γ).(2) Let γ ∈ Γ p,q be any hyperbolic element and γ 0 ∈ Γ p,q a primitive element with γ = ±γ ν 0 for some ν ∈ Z. Then the linking number is given byRemark 4.20. In above, we proved the theorem in S 3 via the case in the lens space. We may also directly discuss the case in S 3 by using automorphic differential forms of degree 1/r studied by Milnor [Mil75, Section 5]. Indeed, we can construct a lift ∆ 1/r p,q : G r \ SL 2 R → C × satisfying ( ∆ 1/r p,q (γ, n)) r = ∆ p,q (γ) for every (γ, n) ∈ SL 2 R. By a similar argument, we may obtainfor γ with the condition of Theorem 4.19 (1). The lift ∆ 1/r p,q equals Tsanov's function ω ∞ (z, dz) in [Tsa13, Lemma 4.16] up to a constant multiple, yielding a homeomorphism G r \ SL 2 R ∼ = S 3 −K p,q [Tsa13, Section 5]. In this subsection, we further introduce another variant Ψ e p,q of the Rademacher symbol as well as define knots corresponding to elliptic and parabolic elements, so that the theorems on linking numbers extends to whole Γ p,q . This symbol is characterized by using an Euler cocycle, which arises as an obstruction to the existence of sections of cycles in the S 1 -bundle T 1 Γ p,q \H ∼ = Γ p,q \ SL 2 R ∼ = L(r, p − 1) − K p,q . Our argument partially justifies Ghys's outlined second proof [Ghy07, Section 3.4] of his theorem. The linking numbers of fibers The singular fibers of the S 1 -bundle corresponding to the elliptic points a = e πi(1−1/p) and b = e πi/q are parametrized asIndeed, they define closed curves byIn addition, for any t ∈ R, we have f a (t)i = a, f b (t)i = b. Bythen c is the image of a holocycle. If γ is hyperbolic, then we further assume thatc is a geodesic. Such c is freely homotopic to a generator of γ in the sense of the orbifold fundamental group. We define the knot C γ as a section of such c. More precisely, in addition to Definition 4.5, we define knots corresponding to elliptic and parabolic elements as follows.Definition 4.21. We put C Sp = C a , C Uq = C b , and C Tp,q = C y discussed in above. In addition, for any g ∈ Γ p,q , we put C ±g −1 S n p g = nC Sp for n = 1, 2, · · · , p − 1 and C ±g −1 U n q g = nC Uq for n = 1, 2, · · · , q − 1. For any g ∈ Γ p,q and n ∈ Z, we put C ±g −1 T n p,q g = nC Tp,q . Definition 4.22. We define the modified Rademacher symbol Ψ e p,q : Γ p,q → Z byWe remark that Ψ e p,q (γ) = ψ p,q (γ) holds if tr γ 2 or γ = S n p (1 n p − 1) or γ = U n q (1 n q − 1). By combining all above, we may conclude the following.Theorem 4.23. For any γ ∈ Γ p,q , the linking number in L(r, p − 1) is given byIn addition, suppose that γ = ±γ ν 0 for a primitive non-elliptic element γ 0 ∈ Γ p,q and ν ∈ Z or γ ∼ ±S n p (1 n p − 1) or γ ∼ ±U n q (1 n q − 1). If C ′ γ is a connected component of h −1 (C γ ) in the sense of Definition 4.14 (2), then the linking number in S 3 is given by lk(C ′ γ , K p,q ) = 1 gcd(r, Ψ e p,q (γ 0 )) Ψ e p,q (γ). An Euler cocycle for Ψ e p,q Let f = f z be a generic fiber given in Section 4.4.1. An Euler cocycle eu : Γ 2 p,q → Z of the S 1 -bundle T 1 Γ p,q \H ∼ = L(r, p − 1) − K p,q is defined by the equalityin H 1 (L(r, p − 1) − K p,q ; Z) for every γ 1 , γ 2 ∈ Γ p,q . Taking the linking numbers with K p,q , we obtainNote that we have H 2 (Γ p,q /{±I}; Z) ∼ = Z/pqZ and C γ = C −γ for any γ ∈ Γ p,q . Let φ : Γ p,q → Z be a unique function satisfying −δφ = pqeu and φ(γ) = φ(−γ) for any γ ∈ Γ p,q . Then for any γ ∈ Γ p,q , we have lk(C γ , K p,q ) = φ(γ)/r. Together with the equality lk(C γ , K p,q ) = Ψ e p,q (γ)/r in Theorem 4.23, we obtain the following.Theorem 4.24. Let eu : Γ 2 p,q → Z denote the Euler cocycle function defined as above. Then the modified Rademacher symbol Ψ e p,q is a unique function satisfying −δΨ e p,q = pqeu and Ψ e p,q (γ) = Ψ e p,q (−γ) for any γ ∈ Γ p,q . Remark 4.25. We may replace Ψ e p,q and eu in Theorem 4.24 by ψ p,q and W by modifying the definition of modular knots for γ's which do not satisfy the condition of Theorem 4.10 (1). In this case, the equalities C γ = C −γ and C γ n = nC γ will be modified according to the formula ψ p,q (−γ) = ψ p,q (γ) + pq sgn(γ).