The Manin-Drinfeld theorem and the rationality of Rademacher symbols

Research Collection

Working Paper

The Manin-Drinfeld theorem and the rationality of Rademacher symbols

Author(s):

Burrin, Claire Publication Date:

2020-12-02 Permanent Link:

https://doi.org/10.3929/ethz-b-000463724

Rights / License:

Creative Commons Attribution 4.0 International

This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use.

ETH Library

arXiv:2012.01147v1 [math.NT] 2 Dec 2020

RADEMACHER SYMBOLS

CLAIRE BURRIN

Abstract. For any noncocompact Fuchsian groupΓ, we show that periods of the canonical differ- ential of the third kind associated to residue divisors of cusps are expressed in terms of Rademacher symbols for Γ — generalizations of periods appearing in the classical theory of modular forms.

This result provides a relation between Rademacher symbols and the famous theorem of Manin and Drinfeld. On this basis, we present a straightforward group-theoretic argument to establish both the rationality of Rademacher symbols and the validity of the Manin–Drinfeld theorem for new families of Fuchsian groups and algebraic curves.

1. Introduction

Let Γ be a noncocompact Fuchsian group. Then Γ\H is an open Riemann surface, and its compactification X can be seen as a smooth algebraic curve. Fixing a base point x ∈X, we have an embedding X ֒→ J(X) into its Jacobian J(X). Manin and Mumford conjectured that for any such embedding, there are only finitely many torsion points in the image. The conjecture was first confirmed by Raynaud [Ray83].

IfΓis a congruence subgroup ofSL₂(Z), Drinfeld [Dri73], generalizing a result of Manin [Man72], provided significant examples of these finitely many torsion points on modular curves; the subgroup C(Γ) < J(X) spanned by the images of the cusps of Γ is contained in J(X)^tor. Their proofs used Hecke operators, limiting the reach of the statement to modular curves. In this note, we observe a new connection between the Manin–Drinfeld theorem and Rademacher symbols, which are ubiquitous class invariants appearing in connection to Dedekind sums and the transformation- theory of the η-function [RG72], class numbers of real quadratic fields [Mey57, Zag75], linking numbers of knots [Ghy07, DIT17], or yet aspects of Atiyah–Bott–Singer index theory [Ati87]. On the basis of this relation, we present a straightforward group-theoretic argument to extend the Manin–Drinfeld theorem to new families of curves.

As is customary, we reframe this discussion in terms of divisors. Recall that a divisor is a formal sum

D=X a_ix_i,

where xi ∈ X, and ai are integers, only finitely many of which are non-zero. The degree of the divisorD is the sum P

a_i, and divisors of degree zero form a group. Let D be a divisor of degree zero. By the Abel–Jacobi theorem, there is an isomorphism between divisor classes (with respect to linear equivalence) of degree zero and the Jacobian J(X) modulo periods. By Riemann–Roch, there is a differential ω_D of the third kind (i.e., a meromorphic differential whose poles are simple and residues are integers) whose residual divisor isD, and this differential can be made unique; see, e.g., Theorem 5.3 in [Lan82]. Then Dis torsion if and only if the periods of ω_D are in2πiQ.

Date: December 3, 2020.

1

Theorem 1. Let Γ be a noncocompact Fuchsian group with h inequivalent cusps a₁, . . . ,a_h. Let D = P

m_i(a_i) to be a divisor of degree zero, and let ω_D be the canonical differential of the third kind associated to the residue divisor D. Then

Z

γ

ω_D = 2πi(m₁Ψa₁(γ) +· · ·+m_hΨa_h(γ)),

where Ψa is the Rademacher symbol for the cusp a.

In particular, if all Rademacher symbols onΓ are rational-valued, then C(Γ)⊂J(X)^tor.

We now describe Rademacher symbols, as they appear in the classical theory of modular forms and elliptic functions. Let q = e(z) = e^2πiz. Dedekind showed that the transformation-theory of theη-function

η(z) =q²⁴¹ Y

n≥1

(1−qⁿ), (1.1)

under the linear fractional action ofγ = ^{a b}_{c d}

∈SL₂(Z), is encoded by the Dedekind symbol

Φ(γ) =





 b

d if c= 0,

a+d

c −12 sign(c)·s(a, c) if c6= 0.

wheres(a, c)are Dedekind sums — arithmetic sums that only depend ona/c, and obey a reciprocity law that makes them easy to compute via the Euclidean algorithm. In his study of Dedekind sums, Rademacher introduced the associated function

Ψ(γ) = Φ(γ)−3 sign(c(a+d)), (1.2)

which has the particular feature of being conjugacy class invariant. In fact, it can be realized as the period

Z

γ

E₂(z)dz= Ψ(γ),

where E₂ is Hecke’s modified Eisenstein series of weight 2; see §3, and [DIT18] for historical refer- ences.

The transformation theory of the η-function can also be deduced from the celebrated first limit formula of Kronecker (see, e.g., [Sie65]), which expresses the constant term in the Laurent expansion of the non-holomorphic Eisenstein series forPSL₂(Z)at its poles= 1. More generally, to each cuspa of a general noncocompact Fuchsian groupΓ, one has an associated such non-holomorphic Eisenstein series, and following Selberg’s work on their analytic properties, one may extend the Kronecker limit formula to this more general setting. In this context, we define in§2 the Rademacher symbol

Ψa: Γ→R,

which is again a conjugacy class invariant. For certain groups, rigorous arithmetic computations suffice to show that the Rademacher symbols are actually rational-valued. This is the case of all principal congruence groups and Hecke congruence groups, following the computations of Dedekind symbols for these groups as in [Dri83], [Tak86] and [Vas96]. Using instead a much more succinct group-theoretic argument in §4, we extend this list as follows.

LetΓ₀(N)⁺ <SL₂(R) be the group obtained by adding all Atkin–Lehner involutions toΓ₀(N).

(We refer the reader to §4 for an explicit parametrization.) This family of groups is of particular

interest due to the following theorem of Helling [Hel66]; every subgroup of SL2(R) that is com- mensurable to SL₂(Z) (i.e., arithmetic) is conjugate to a subgroup of Γ₀(N)⁺ for some squarefree N. Moreover, each maximal noncocompact arithmetic group (under inclusion) is a conjugate of Γ₀(N)⁺ for some squarefree N. (The converse does not hold; see [Bai87].) When N is squarefree, Γ₀(N)⁺has a single cusp (see [JST14]), and a further notable feature is that the 44 groupsΓ₀(N)⁺ of genus 0 appear in connection to monstruous moonshine (see [Cum04] and references therein).

Theorem 2. Rademacher symbols are rational for all genus 0 noncocompact Fuchsian groups, for all Γ₀(N)⁺, for all noncocompact Fuchsian triangle groups, and for their normal subgroups of finite index.

Finite-index normal subgroups of Fuchsian triangle groups are precisely the uniformizing groups of Galois Belyi curves, i.e. curves that admit a Galois Belyi map, following the terminology of [CV19] — sometimes also called curves with many automorphisms, triangle curves, or quasiplatonic surfaces.

There are only finitely many such curves for each given genus, but these include much studied examples. For instance, the Fermat curves, for which Manin–Drinfeld was already established by Rohrlich [Roh77], who also studied the structure of the divisor class group generated by the cusps.

Together,Theorem 1andTheorem 2extend the statement of Manin–Drinfeld to Galois Belyi curves.

The general situation is not so simple: Rohrlich observed that for some non-congruence subgroups of Γ(2), C(Γ) must be infinite; see p. 198 in [KL81]. We can therefore not expect Rademacher symbols to be rational for general Γ.

1.1. Acknowledgments. This note was inspired by the beautiful paper of Murty and Ramakr- ishnan [MR87] on a similar connection between the Manin–Drinfeld theorem and generalized Ra- manujan sums. The author thanks Jay Jorgenson for many discussions on the connections between Manin–Drinfeld and Dedekind symbols of various types.

2. Rademacher symbols for cusps

LetΓdenote a noncocompact Fuchsian group. The classical monographs of Shimura [Shi71] and Kubota [Kub73] and the more recent book of Iwaniec [Iwa02] serve as references for facts cited in this section on Fuchsian groups and Eisenstein series.

2.1. Cusps, Eisenstein series. Let a be a cusp for Γ, and let Γa be the stabilizer subgroup of a, i.e., Γa = {γ ∈ Γ : γa = a}. The group Γa is infinite cyclic. We will denote by γa the cyclic generator of Γa. We say that two cusps a,bare equivalent ifb=γa for someγ ∈Γ. Ifaand bare equivalent, then Γa= Γb.

We may choose a scaling transformation σa∈PSL(2,R) such that σa(∞) =a and σa⁻¹Γaσa=±

1 Z 1

.

Such a choice ofσa is only unique up to right multiplication by an element of± ¹0 1^R

. Conjugating Γ byσa provides a group with a cusp at infinity of width 1, and facilitates computation.

The group Γ acts discontinuously on the upper half-planeH by fractional linear transformation

± ^{a b}_{c d}

:z7→ ^az+b_cz+d. For each z∈H, the Eisenstein series for the cusp ais defined by Ea(z, s) = X

γ∈Γa\Γ

Im(σa⁻¹γz)^s,

where sis a complex parameter controlling the convergence of the infinite series; the seriesEa(z, s) converges absolutely and uniformly on compact subsets when Re(s) >1. Beyond this half-plane, Eisenstein series admit a meromorphic continuation to the whole complex plane, as famously proved by Selberg. The definition of Ea(z, s) does not depend on the particular choice of scaling σa. Moreover, equivalent cusps yield identical Eisenstein series.

To understand the behavior of the Eisenstein seriesEaat the (not necessarily distinct) cuspb of Γ, one considers the seriesEa(σbz, s), for which we have the relation

Ea(σbz, s) =Ea(γbσbz, s) =Ea(σb(z+ 1), s) and the resulting Fourier expansion is explicitly given by

Ea(σbz, s) =δaby^s+

√πΓ(s−1/2)

Γ(s) ϕab(s)y^1−s+X

n6=0

2π^s|n|^s−1/2Γ(s)⁻¹√yK_s−1/2(2π|n|y)ϕab(n, s)e(nx),

where δab = 1 if the cusps a and b are in the same Γ-orbit, and 0 otherwise, where Γ(s) denotes the classical Γ-function, whereK_s(z) is the K-Bessel function (or, modified Bessel function of the second kind), where e(z) =e^2πiz, and whereϕab(s) andϕab(n, s) are the Dirichlet series

ϕab(s) =X

c>0

c^−2s#{d∈[0, c) : (^{∗ ∗}_{c d})∈σ_a⁻¹Γσb},

ϕab(n, s) =X

c>0

c^−2s X

0≤d<c

(^{∗ ∗}_{c d})∈σ⁻a¹Γσb

e

nd c

.

These Dirichlet series do not depend on a particular representative for the cuspsa,b, butϕab(n, s) depends on the particular choice of the scaling transformation σb up to a unitary multiplicative factor. Indeed, if we replaceσb byσbn_x,n_x =±(¹_{0 1}^x), thenϕab(n, s)is replaced by ϕab(n, s)e(nx).

Both Dirichlet series are absolutely convergent onRe(s)>1. On the vertical lineRe(s) = 1, the Eisenstein series are holomorphic, save for a simple pole ats= 1. Examining the local behavior of its Fourier coefficients, we note that the K-Bessel function is entire as a function of s — in fact, at s = 1, we have 2(|n|y)^1/2K_1/2(2π|n|y)e(nx) = e^−2π|n|ye(nx), see [Iwa02, p. 227] —, and that the Dirichlet series ϕab(n, s) are holomorphic along the vertical lineRe(s) = 1. At s= 1,ϕab(n,1) is real valued, hence ϕab(n,1) =ϕab(−n,1), and satisfies the estimate |ϕab(n,1)| ≪n^1+ǫ, for any ǫ > 0; see [JO05, Theorem 1.1]. Finally, the Dirichlet series defining ϕab(s) has a simple pole at s= 1 of residue(πV)⁻¹, whereV denotes the hyperbolic volume of any fundamental domain forΓ.

Thusϕab(s) has a Laurent expansion ats= 1 of the form ϕab(s) = (πV)⁻¹

s−1 +X

n≥0

c_n(s−1)ⁿ.

2.2. Kronecker’s first limit formula. A modern extension of Kronecker’s first limit formula (see [Gol73, Theorem 3-1]) gives the constant coefficient in the Laurent series for Ea(σbz, s) at the simple pole at s= 1. Using the Fourier expansion of the Eisenstein series and letting s→1⁺, we immediately obtain the corresponding formal Laurent series at s= 1, given by

slim→1⁺

Ea(σbz, s)− V⁻¹ s−1

= c0−V⁻¹logy+δaby+X

n>0

πϕab(n,1)e(nz) +X

n<0

πϕab(n,1)e(nz)

= c₀−V⁻¹logy−Re δabiz−2πX

n>0

ϕab(n,1)e(nz)

! .

We denote the term in parenthesis by fa,σb(z) and record that fa,σbnx(z) =fa,σb(nxz)−δabix. As a function of z,fa,σb is holomorphic; this establishes the validity of the limit formula.

Specializing to Γ = PSL₂(Z), we recover the classical first limit formula of Kronecker as follows.

We have

E(z, s) = 1 2

X

c,d∈Z (c,d)=1

y^s

|cz+d|^2s, ϕ(s) =X

c≥1

φ(c) c^2s ,

ϕ(n, s) =X

c≥1

c^−2s Xc (c,d)=1d=1

e

nd c

=X

c≥1

1 c^2s

X

δ|(c,n)

δµc δ

= 1 ζ(2s)

X

δ|n

1 δ,

where φ is Euler’s totient function, and we have used Kluyven’s identity for Ramanujan sums in the last line. Using that ζ(2) = ^π₆²,

f(z) = 12 π



πiz

12 − X

m,n>0

q^mn m



,

and the term in parenthesis is a branch of the logarithm of Dedekind’sη-function, defined by (1.1).

2.3. Dedekind and Rademacher symbols. We introduce two auxiliary functions on Γ; the inner automorphism τ_σb(γ) =σ⁻_b¹γσb, and the automorphic cocyclej(γ, z) =cz+d. The latter is a multiplicative cocycle: for any pair γ1, γ2 ∈Γ, we have j(γ1γ2, z) =j(γ1, γ2z)j(γ2, z). The limit formula derived above implies the relation

Refa,σb(τb(γ)z) =V⁻¹ln|j(τb(γ), z)|²+Refa,σb(z)

for eachγ ∈Γ. We chooselogz to be the principal branch of logarithm, i.e.,logz= ln|z|+iarg(z) with arg(z) ∈ (−π, π]. Then log(−j(γ, z)²) is well defined, and Re sign(c)²log(−j(γ, z)²)

= ln|j(γ, z)|². Therefore, the holomorphic function

Fa,σb(γ;z) :=fa,σb(τ_σb(γ)z)−fa,σb(z)−V⁻¹sign(c_τσb(γ))²log −j(τ_σb(γ), z)²

(2.1) has trivial real part, and thus by the Open Mapping Theorem, it must be a constant function ofz, i.e., Fa,σb(γ;z) =Fa,σb(γ). As a result, it is also independent of the particular choice of the scaling transformation σb, since

Fa,σbn^x(γ) =fa,σb(τ_σb(γ)n_xz)−fa,σb(n_xz)−V⁻¹sign(c_τσb(γ))²log −j(τ_σb(γ), n_xz)²

=Fa,σb(γ).

We henceforth denote this functionFab. We can now define the Dedekind and Rademacher symbols attached to the cusp a. Set

Φab(γ) :=−iFab(γ), Φa(γ) := Φaa(γ) and

Ψa(γ) := Φa(γ)−πV⁻¹sign(c(a+d)).

with ^{a b}_{c d}

= τ_σa(γ). When Γ = PSL(2,Z), we recover the classical Dedekind and Rademacher symbols Φ and Ψ; see Chapters 4A-C in [RG72]. A detailed study of the properties of (a slight modification of) the Dedekind symbolsΦab appears in [JOS20]. We only record here that, following the analysis in pp. 52–53 of [RG72],

Φa(γ₁γ₂) = Φa(γ₁) + Φa(γ₂)−πV⁻¹sign(c₁c₂c₃) (2.2) for any γ₁,γ₂∈Γ, with (_c^{∗ ∗}1∗) =τ_σa(γ₁),(_c^{∗ ∗}2∗) =τ_σa(γ₂), and (_c^{∗ ∗}3 ∗) =τ_σa(γ₁γ₂).

Proposition 2.1. Rademacher symbols are rational on elliptic and parabolic motions.

In particular, Rademacher symbols are rational on genus 0 noncocompact Fuchsian groups.

Proof. Letγ be a parabolic element. Recall thatΓbis an infinite cyclic group and denote its genera- tor byγb. Letγ =γ_b^mfor somem∈N. We first observe thatΦab(γ) =−i(fa,σb(z+m)−fa,σb(z)) = δab·m.

SinceEa(σaz, s) =Ea(σbz^′, s) withz^′ =σ⁻_b¹σaz, the Kronecker limit formula implies the relation fa,σb(z^′)−fa,σa(z) =V⁻¹log −j(σ_b⁻¹σa, z)²

+constant.

Then

Φa(γ) =−i fa,σ^a(τ_σa(γ)z)−fa,σ^a(z)−V⁻¹log −j(τ_σa(γ), z)²

=−i fa,σb(z^′+m)−V⁻¹log(−j(σ⁻_b¹σa, τ_σa(γb)z)²

−fa,σb(z^′) +V⁻¹log −j(σ_b⁻¹σa, z)²

−V⁻¹log −j(τσa(γ), z)² )

= Φab(γ) + 2iV⁻¹

logj(σ⁻_b¹σa, τ_σa(γ)z) + logj(τ_σa(γ), z)−logj(σ_b⁻¹σa, z) +iπ

2Aab(γ)

=δab·m+ 2iV⁻¹

2πiBab(γ) +iπ

2Aab(γ) ,

where Aab and Bab are integer-valued. Looking at the Gauss–Bonnet formula for V (see, e.g., [Iwa02, p. 46]), we have πV⁻¹ ∈Q and this concludes. Let now γ be an elliptic element of order m. Applying (2.2) recursively, we have

0 = Φa(γ^m) =mΦa(γ)−πV⁻¹ Xm k=2

sign(c_γc_γ^k−1c_γ^k)

and this shows that Φa(γ)∈ Q. Since Fuchsian groups are finitely generated and the transforma- tion formula (2.2) for Φa depends only on the rational constant πV⁻¹, we conclude that genus 0 noncocompact Fuchsian groups have rational-valued Rademacher symbols.

In the next section, we show that Rademacher symbols on hyperbolic elements can be expressed as the periods of modified Eisenstein series of weight 2; seeLemma 3.1 below.

3. Proof of Theorem 1 For each cusp a, consider the modified Eisenstein series

E2,a(z, s) := 2i∂

∂zEa(z, s) = ∂

∂y +i ∂

∂x

Ea(z, s).

Since the residue of Ea(z, s) at the simple pole at s= 1 is constant in z, this modified Eisenstein series is regular at s= 1, and we setE_2,a(z) :=E_2,a(z,1). Explicitly, we have

E_2,a(z) = lim

s→1⁺sy^s−1 X

γ∈Γ^a\Γ

j(σ⁻a¹γ, z)⁻²|j(σa⁻¹γ, z)|^−2s+2.

Hence this is a real-analytic Eisenstein series of weight 2; for principal congruence groups, this is Hecke’s construction. The Kronecker limit formula for Ea(σbz, s) yields the Fourier expansion of

E2,a(σbz), which is then given by

E2,a(σbz)σb^′(z) = 2i∂

∂z c0−V⁻¹logy+δaby+X

n>0

πϕab(n,1)e(nz) +X

n<0

πϕab(n,1)e(nz)

!

=−V⁻¹

y +δab−4π²X

n>0

nϕab(n,1)e(nz)

=−V⁻¹ y +1

i d

dzfa,σb(z).

The relation (2.2) implies that Ψa(γ⁻¹) = −Ψa(γ) and Ψa(−γ) = Ψa(γ). Thus up to replacing γ = ^{a b}_{c d}

by ±γ^±1, we may assume thatc >0 anda+d >0.

Lemma 3.1. Let γ ∈Γ be an element of positive trace. Then Z

γ

E_2,a(z)dz = Ψa(γ).

Proof. Fix a base point z₀ ∈H. Using the above Fourier expansion, Z γz0

z0

E_2,a(z)dz=

Z σa⁻¹γ(z0) σ⁻a¹(z0)

E_2,a(σaz)σ_a^′(z)dz

=−V⁻¹

Z σ⁻a¹γ(z0) σ⁻a¹(z0)

dz y +1

i f_σa(τ_σa(γ)σa⁻¹(z₀))−f_σa(σ⁻a¹(z₀)) .

For any g= (^{∗ ∗}_{c d})∈SL₂(R), we have dz

y ◦g− dz

y = |cz+d|²−(cz+d)²

(cz+d)²y dz=− 2ic

cz+ddz =−2i dlogj(g, z) and this shows that

Z τ^σa(γ)σa⁻¹(z0) σa⁻¹(z0)

dz

y =−2ilogj(τσa(γ), σ⁻a¹(z0)).

On the other hand,

f_σa(τ_σa(γ)σa⁻¹(z₀))−f_σa(σ⁻a¹(z₀)) =iΦa(γ) +V⁻¹sign(c_τσa(γ))²log −j(τ_σa(γ), σ⁻a¹(z₀))² Ifc_τσa(γ) 6= 0, then

log −j(τ_σa(γ), σa⁻¹(z₀))²

= 2 log

c_τσa(γ)z+d_τσa(γ)

isign(c_τσa(γ))

= 2 log(c_τσa(γ)z+d_τσa(γ))−iπsign(c_τσa(γ)) and we conclude that Z γz0

z0

E_2,a(z)dz= Φa(γ)−πV⁻¹sign(c_τσa(γ)),

which coincides with the definition of the Rademacher symbol Ψa(γ) for γ of positive trace.

Let a₁,a₂, . . . ,a_h denote the inequivalent cusps of Γ. For each integer tuple m = (m₁, . . . , m_h) such that P

m_i = 0,define

E_2,m(z) :=

Xh i=1

m_iE_2,ai(z).

ThenE2,m(z) has the Fourier expansion

E_2,m(σbz)j(σb, z)⁻² =mb−4π² Xh

i=1

m_iX

n≥1

nϕaib(n)e(nz),

from which it can be seen that E_2,m(z) is a holomorphic modular form of weight 2. In particular, E_2,m induces a holomorphic differential on X and we conclude our proof of Theorem 1 with the following result of Scholl; see Proposition 2 in [Sch86].

Proposition 3.2. Let m= (m₁, . . . , m_h). In the notation above, 2πiE_2,m(z)dz =ω_D

which is the canonical differential of the third kind associated to the residue divisor D=P

m_i(a_i).

4. Proof of Theorem 2

4.1. Three group-theoretic propositions and their applications. LetT =R/Z denote the torus seen as an additive group. For any groupG, let Gb= Hom(G,T) be its character group.

Definition 4.1. We say that χ∈Gb istorsion if its image is finite and rational.

Proposition 4.2. If |Gb|<∞ then each character on Gis torsion.

Proof. Suppose that there is a character χ ∈ Gb such that χ(g₀) = α is irrational. Consider the family {τ_n}n∈N ⊂Gb given by τ_n(g) :=χ(gⁿ). Then τ_n(g₀) =nα (mod 1), and since the sequence (nα)_n∈^N becomes equidistributed mod 1, we conclude that there exists infinitely many distinct

charactersτ_ninG.b

LetΨa: Γ→Tdenote the composition of Ψa with the canonical projection R→T.

Proposition 4.3. There exists a positive integer n≥1 such that Ψa,n :=nΨa andΨa,n ∈Γ.b Moreover, each Rademacher symbol Ψa is rational if and only if Ψa,n is torsion.

Proof. Following (2.2), Φa fails to be a homomorphism by a defect that is an integer multiple of πV⁻¹. By definition, the same holds true ofΨa. Looking at the Gauss–Bonnet formula for V (see, e.g., [Iwa02, p. 46]), there exists a smallest positive integer nsuch that nπV⁻¹ ∈N. Hence for any γ,γ^′∈Γ,nΨa(γγ^′)≡nΨa(γ) +nΨa(γ^′) (mod1).

Proposition 4.4. Let N ⊳ G be a normal subgroup of finite index. Then we have |Nb|<∞ if and only if |Gb|<∞.

Proof. Let H be the group H = G/N. Since H is finite, we have |Hb| < ∞. The short exact sequencee−→N −→G−→H −→einduces the further short exact sequence0−→Hb −→Gb−→

Nb −→0.

Together, these three propositions provide straightforward algebraic proofs of both the rationality of the classical Rademacher symbol defined by (1.2), and of the Manin–Drinfeld theorem.

Corollary 4.5. The classical Rademacher symbol is a rational-valued function.

Proof. Indeed, G:= PSL2(Z)∼=C2∗C3 and hence Gb= Hom(G,¹₆Z∩T) is finite.

Corollary 4.6. If Γ is a congruence subgroup of PSL₂(Z), then C(Γ)⊂J(X)^tor.

Proof. A discrete subgroupΓ<SL2(Z)is congruence if it contains a principal congruence subgroup Γ(N) and thatΓ(N)is a normal subgroup of SL₂(Z). ApplyingProposition 4.4 twice, we conclude that |Γb|<∞. All Rademacher symbols on Γ are therefore rational-valued, and by Theorem 1, for any residue divisorD=P

m_i(a_i), the periods of ω_D lie in 2πiQ.

In the rest of this section, we build on this approach to extend both the rationality of Rademacher symbols and the result of Manin and Drinfeld to other groups that are not easily accessible to direct rigorous computation.

4.2. Noncocompact Fuchsian triangle groups. Fuchsian triangle groups are the orientation- preserving isometry groups of hyperbolic triangles T_a,b,c, i.e., triangles determined by angles ^π_a, ^π_b,

π

c such that

1

a+¹_b +¹_c < 1.

Ifτ_a,τ_b,τ_c denote the reflections across the sides ofT_a,b,c, then △a,b,c:= Isom⁺(T_a,b,c)is generated by γ_a=τ_bτ_c,γ_b =τ_cτ_a,γ_c =τ_aτ_b. More precisely,

△a,b,c=D

γ_a, γ_b, γ_c|γ_a^a=γ_b^b =γ_c^c =γ_aγ_bγ_cE

<PSL₂(R),

and △a,b,c\H is an open Riemann surface of genus zero. It is noncocompact if at least one of a, b, c =∞. Two triangle groups of the same type (a, b, c) are conjugate in PSL₂(R); see [Tak77].

Hence, a noncocompact Fuchsian triangle group is isomorphic to either△∞,∞,∞,△a,∞,∞, or△a,b,∞, whereby △∞,∞,∞∼= Γ(2), and△a,b,∞∼=C_a∗C_b. Following the discussion above, we conclude that in the latter two cases, the corresponding character groups are finite. The following proposition implies that this is also true of triangle groups of type(a,∞,∞).

Proposition 4.7. The group△a,∞,∞, is isomorphic to an index 2 subgroup of △2,a,∞. Proof. A standard realization of △a,∞,∞ is given by the set of generators

γ₁=

−2 cos(^π_a) −1

1 0

, γ₂ =

0 1

−1 2

, γ₃=

1 2(cos(^π_a + 1)

0 1

,

whereby γ₁^a= 1in PSL₂(R). Let µ= 2(1 + cos(^π_a)). Conjugating these generators by √µ1/√µ 0 1/√µ

, we have

e γ₁ =

1−µ −1

µ 1

, eγ₂ =

1 0

−µ 1

, eγ₃ = 1 1

0 1

.

With respect to this realization, the triangle group has two inequivalent cusps at 0 and ∞. The involution ω = _{0 −1/}√µ

õ 0

normalizes the conjugated group; indeed, we have ω⁻¹eγ₂ω =eγ₃. The resulting supergroup

Γ^′ =△^a,∞,∞∪ω△^a,∞,∞

then has a single cusp at ∞ and two elliptic generators, given by ω and eγ₁, of orders 2 and a,

respectively.

Together with Proposition 4.4, we conclude that if Γ is a normal subgroup of finite index of a noncocompact triangle group, then Γb is finite. In particular, all Rademacher symbols on Γ are rational-valued.

4.3. The groups Γ0(N)⁺. The groups Γ0(N)⁺ are obtained from Γ0(N) by adding all (finitely many) Atkin–Lehner involutions and normalizing to matrices of determinant 1 by taking the quotient by a positive scalar. Explicitly, one has

Γ₀(N)⁺ =

e^−1/2 a b

c d

∈SL₂(R) :a, b, c, d, e∈Z, e||N, e|a, d, N |c

,

where e || N signifies that e | N and (e,^N_e) = 1. It is easily seen that the groups Γ₀(N)⁺ are commensurable to SL₂(Z) but not necessarily conjugate conjugate to a subgroup of SL₂(Z).

Moreover,Γ₀(N)⁺ containsΓ₀(N)as a normal subgroup of index2^ω(N), whereω(N)is the number of distinct prime divisors of N, and as such its character group is finite; the Rademacher symbols on Γ₀(N)⁺ are rational.

References

[Ati87] Michael Atiyah. The logarithm of the Dedekindη-function.Math. Ann., 278(1-4):335–380, 1987.

[Bai87] Walter L. Baily, Jr. Maximal arithmetic groups: a correction: “On the theory of Hilbert modular functions.

I. Arithmetic groups and Eisenstein series” [J. Algebra90(1984), no. 2, 567–605; MR0760029 (85j:11051)].

J. Algebra, 107(1):183–186, 1987.

[Cum04] C. J. Cummins. Congruence subgroups of groups commensurable withPSL(2,Z)of genus 0 and 1.Experi- ment. Math., 13(3):361–382, 2004.

[CV19] Pete L. Clark and John Voight. Algebraic curves uniformized by congruence subgroups of triangle groups.

Trans. Amer. Math. Soc., 371(1):33–82, 2019.

[DIT17] W. Duke, Ö. Imamo¯glu, and Á. Tóth. Modular cocycles and linking numbers.Duke Math. J., 166(6):1179–

1210, 2017.

[DIT18] W. Duke, Ö. Imamo¯glu, and Á. Tóth. Kronecker’s first limit formula, revisited.Res. Math. Sci., 5(2):Paper No. 20, 21, 2018.

[Dri73] V. G. Drinfeld. Two theorems on modular curves.Funkcional. Anal. i Priložen., 7(2):83–84, 1973.

[Dri83] Y. Driencourt. Sur les sommes de Dedekind attachées aux groupes de congruence.Bull. Soc. Math. France, 111(4):373–419, 1983.

[Ghy07] Étienne Ghys. Knots and dynamics. InInternational Congress of Mathematicians. Vol. I, pages 247–277.

Eur. Math. Soc., Zürich, 2007.

[Gol73] Larry Joel Goldstein. Dedekind sums for a Fuchsian group. I.Nagoya Math. J., 50:21–47, 1973.

[Hel66] Heinz Helling. Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe.Math. Z., 92:269–

280, 1966.

[Iwa02] Henryk Iwaniec. Spectral methods of automorphic forms, volume 53 of Graduate Studies in Mathematics.

American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, second edi- tion, 2002.

[JO05] Jay Jorgenson and Cormac O’Sullivan. Convolution Dirichlet series and a Kronecker limit formula for second-order Eisenstein series.Nagoya Math. J., 179:47–102, 2005.

[JOS20] Jay Jorgenson, Cormac O’Sullivan, and Lejla Smajlović. Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series.Res. Number Theory, 6(2):Paper No. 22, 42, 2020.

[JST14] Jay Jorgenson, Lejla Smajlović, and Holger Then. On the distribution of eigenvalues of Maass forms on certain moonshine groups.Math. Comp., 83(290):3039–3070, 2014.

[KL81] Daniel S. Kubert and Serge Lang.Modular units, volume 244 of Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Science]. Springer-Verlag, New York-Berlin, 1981.

[Kub73] Tomio Kubota.Elementary theory of Eisenstein series. Kodansha Ltd., Tokyo; Halsted Press [John Wiley

& Sons], New York-London-Sydney, 1973.

[Lan82] Serge Lang.Introduction to algebraic and abelian functions, volume 89 ofGraduate Texts in Mathematics.

Springer-Verlag, New York-Berlin, second edition, 1982.

[Man72] Ju. I. Manin. Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR Ser. Mat., 36:19–66, 1972.

[Mey57] Curt Meyer.Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern. Akademie- Verlag, Berlin, 1957.

[MR87] V. Kumar Murty and Dinakar Ramakrishnan. The Manin-Drinfeld theorem and Ramanujan sums. Proc.

Indian Acad. Sci. Math. Sci., 97(1-3):251–262 (1988), 1987.

[Ray83] M. Raynaud. Courbes sur une variété abélienne et points de torsion.Invent. Math., 71(1):207–233, 1983.

[RG72] Hans Rademacher and Emil Grosswald.Dedekind sums. The Mathematical Association of America, Wash- ington, D.C., 1972. The Carus Mathematical Monographs, No. 16.

[Roh77] David E. Rohrlich. Points at infinity on the Fermat curves.Invent. Math., 39(2):95–127, 1977.

[Sch86] A. J. Scholl. Fourier coefficients of Eisenstein series on noncongruence subgroups.Math. Proc. Cambridge Philos. Soc., 99(1):11–17, 1986.

[Shi71] Goro Shimura.Introduction to the arithmetic theory of automorphic functions. Publications of the Mathe- matical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1.

[Sie65] Carl Ludwig Siegel.Lectures on advanced analytic number theory. Notes by S. Raghavan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 23. Tata Institute of Fundamental Research, Bombay, 1965.

[Tak77] Kisao Takeuchi. Arithmetic triangle groups.J. Math. Soc. Japan, 29(1):91–106, 1977.

[Tak86] Ichiro Takada. Dedekind sums ofΓ(N).Japan. J. Math. (N.S.), 12(2):401–411, 1986.

[Vas96] I Vassileva.Dedekind eta function, Kronecker limit formula and Dedekind sum for the Hecke group. PhD thesis, University of Massachussets, Amherst, 1996.

[Zag75] Don Zagier. Nombres de classes et fractions continues. InJournées Arithmétiques de Bordeaux (Conf., Univ.

Bordeaux, Bordeaux, 1974), pages 81–97. Astérisque, No. 24–25. 1975.

ETH Zürich, D-MATH, Zurich, Switzerland Email address: claire.burrin@math.ethz.ch