AUTOMORPHIC FORMS FOR SUBGROUPS OF THE

MODULAR GROUP. II: GROUPS CONTAINING CONGRUENCE SUBGROUPS

HERSHEL M. FARKAS AND IRWIN KRA

1. Introduction

This paper is a continuation of the study begun in [?]. One of our main aims is to investigate the function theory of surfaces uniformized by subgroups of the modular group Γ = PSL(2, ℤ); in particular, by the principal congruence groups Γ(k) and subgroups that contain them. The most important among these are the subgroups G(k); defined as the groups formed by adjoining the translation (in ℂ̂) τ ↦ τ + 1 to the principal congruence subgroup of level k, and the Hecke subgroups

Γ_{o}(k) which consist of the elements [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ Γ with c ≡ 0 mod k.

The groups Γ(k) play a prominent role in our work, in part because of the natural identification of the punctures on ℍ^{2}/Γ(k) with classes of characteristics X_{o}(k), a subset of the set X(k) of characteristic classes invariant under Γ(k). In [?] we developed a theory of rational charac-

teristics [ \frac{m}{k} ; \frac{m′}{k} ] (here m, m′ and k are positive integers with k a positive

odd prime) for theta constants, and used the theory to construct modular forms for the level k congruence subgroups of the modular group. The modular forms were then used to construct modular functions for Γ(3) and thus we obtained among other things, a holomorphic homeomorphism of ℍ^{2}/Γ(3) onto a four times punctured sphere with cross ratio ϵ = exp(−πı/3). See §8.2. In fact, we obtained several such maps and as a consequence some new theta constant identities. In particular, we obtained the identity

θ^{3} [ 1^{\frac{1}{3}} ] = θ^{3} [ \frac{1}{3} \frac{1}{3} ] + θ^{3} [ \frac{1}{3} \frac{5}{3} ] .

Research by HMF supported in part by the Paul and Gabriella Rosenbaum Foundation and the Edmund Landau Center for Research in Mathematical Analysis with funding from the Minerva Foundation. Research by IK supported in part by NSF Grant DMS 9500557.

This identity, while not difficult to prove after one discovers its existence, raises the question of whether there are identities of this type for all primes k. This question was settled in the affirmative by Farkas and Kopeliovich. They show that for any odd integer k, there is an identity

∑_{l=0}^{k−1}(−1)^{l}θ^{k} [ \frac{1}{k} ; \frac{2l+1}{k} ] = 0.

The proof (see [?] and [?]) uses the residue theorem. The method used has a wide range of applicability and is sufficiently powerful to recapture many classical identities.

An interesting phenomenon that occurs in our work on the identities arising from the set X(k) is the appearence of new characteristics. In fact, in [?] we already considered, for k = 2 and k = 3, a complementary set Y (k) to X(k) that was used to construct certain modular (not necessarily cusp) forms.

In this paper we show how these classes of characteristics arise in a natural and formal way from the theory of congruence subgroups of the modular group for composite level k. We obtain a correspondence between punctures on the Riemann surface ℍ^{2}/Γ(k) and a certain set of classes of characteristics which, as remarked earlier, we call X_{o}(k). We treat the cases k = 4 and 6 in great detail. We obtain, as a result of the k = 4 theory, an identity which also follows from [?]. The current approach explains the identity from the point of view of covering maps. For the k = 6 case (where the Riemann surface involved is a twelve punctured torus) we obtain a more or less complete description of the function theory and identities of a different form.

We believe that our representation of punctures on ℍ^{2}/Γ(k) as classes of characteristics is not only a useful tool for the construction of modular forms. It is in sense a very natural identification that should have other applications. In order to substantiate this claim, we show (as a consequence of our work on automorphic forms) that there are a number of simple formulae for the genus (equivalently, the number of punctures) of the surface ℍ^{2}/Γ(k) in terms of the characteristics classes corresponding to the punctures.

This paper together with its predecessor lay the foundation for an ongoing study (see [?], [?] and [?]). It contains a number of results on uniformizations of surfaces and on theta constant identities of independent interest. We also include (in §9) an application of theta fuctions to the study of conformalmappings. Because this paper is a direct continuation of [?], we employ the notation introduced there even though it does not always conform with the notation established by the group and number theorists who have done an enormous amount of work on the modular group and its subgroups. The notation and language we established in [?] is consistent with the language and terms used by researchers, mostly complex anaslysts, in the theory of Fuchsian and Kleinian groups. As our background is in the latter group, we have tried to write in a style and language more familiar to them. We trust that this will not make it excessively difficult for the readers whose background is group theory or number theory, since we hope that they will be interested in our results.

An example of the confusion which may arise is in our calling certain classes of characteristics related to the subgroup Γ(k) by the names X(k), X_{o}(k). These symbols are traditionally used to denote the Riemann surfaces obatined by identifying the upper half plane under the groups Γ(k), Γ_{o}(k). Another deviation from the traditions established by the group and number theorists is calling a certain subgroup of the modular group G(k) rather than Γ_{1}(k). Less serious is our using ℍ^{2} to denote the upper half plane rather than the more traditional ℍ. We trust that the context will make everything clear and not cause too much confusion. We assume that the reader is familiar with content (especially the notation) of [?].

2. Three families of groups

As in [?], we are studying subgroups of the modular group Γ = PSL(2, ℤ) acting as Möbius transformations on the upper half plane

ℍ^{2} = {τ ∈ ℂ; ℑτ > 0}.

The generators of SL(2, ℤ), in addition to −I (we use the same symbols for a matrix and the Möbius transformation it induces), are

A = [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] and B = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] .

2.1. Let G be a subgroup of finite index μ = μ(G) of Γ. Alternately, we may describe G as a finitely generated Fuchsian group of the first kind contained in Γ. The index is given by the formula

μ = \frac{Area(ℍ^{2}/G)}{Area(ℍ^{2}/Γ)} = 6 (2p − 2 + ∑_{j=1}^{n} (1 − \frac{1}{μ_{j}} )) ,

where (p, n; μ_{1}, ..., μ_{n}) is the signature of G. Since Γ contains elliptic elements of orders 2 and 3 only, it makes sense to let:

ν_{2} = ν_{2}(G) be the number of G-inequivalent elliptic fixed points (in ℍ^{2}) of order 2 (equivalently the number of G-conjugacy classes of elliptic subgroups of order 2), ν_{3} = ν_{3}(G) be the number of G-inequivalent elliptic fixed points (in ℍ^{2}) of order 3, and ν_{∞} = ν_{∞}(G) be the number of G-inequivalent parabolic fixed points (in ℚ ∪ {∞}).

Obviously, ν_{i} is the number of conjugacy classes in G of elliptic subgroups of order i = 2, 3, and ν_{∞} is the number of conjugacy classes in G of maximal (cyclic) parabolic subgroups. While ν_{2} and ν_{3} are only nonnegative, ν_{∞} must be positive (because of the finite index assumption).

Slightly at variance with [?], we will call the quadruple {μ; ν_{2}, ν_{3}, ν_{∞}}, the branch schema for G. This quadruple is a topological invariant for the orbifold ℍ^{2}/G; its genus p = p(G) is computed as (see, for example, [?, Proposition 1.40])

p = 1 + \frac{μ}{12} − \frac{ν_{2}}{4} − \frac{ν_{3}}{3} − \frac{ν_{∞}}{2} .

The above formula can be obtained from Riemann-Hurwitz applied to the natural covering ℍ^{2}/G → ℍ^{2}/Γ. It can also be obtained immediately from the formula for the index given at the beginning of this subsection. The signature of G is (p, ν_{2}+ν_{3}+ν_{∞}; 2, ..., 2, 3, ..., 3, ∞, ..., ∞) with i listed ν_{i} times for i = 2, 3 and ∞. The canonical projection

P = P (G) : ℍ^{2} → ℍ^{2}/G

extends to the set of fixed points, ℚ_{𝔾} ⊂ ℚ∪{∞}, of parabolic elements of G; we will use the same symbol P to denote the extension, and we will often write P_{x} for P (x) with x ∈ ℍ^{2} ∪ ℚ_{G}. The compactification of ℍ^{2}/G obtained by filling in the punctures will be denoted by \overline{ℍ^{2}/G}. We thus also view

P : ℍ^{2} ∪ ℚ_{𝔾} → \overline{ℍ^{⊭}/𝔾}.

2.2. The principal congruence subgroups Γ(k). Throughout this paper, k is a positive integer; k ∈ ℤ^{+}. As in our previous paper [?], we let Γ(k) denote the level k principal congruence subgroup^{1} of Γ; it

consists of Möbius transformations represented by matrices [ \begin{matrix}a & b \\ c & d \end{matrix} ]

with

a ∈ ℤ, a ≡ 1 mod k, b ∈ ℤ, b ≡ 0 mod k,
c ∈ ℤ, c ≡ 0 mod k, d ∈ ℤ, d ≡ 1 mod k,

1Thus Γ(1) = Γ.

and ad − bc = 1.

Let red denote the canonical reduction map

red : ℤ → ℤ_{k} = ℤ/kℤ.

The map red induces a surjective homomorphism

red : SL(2, ℤ) → SL(2, ℤ_{k}).

For k > 2, Γ(k) is isomorphic to the kernel of this map; while Γ(2) is isomorphic to the kernel of this map factored by {±I}. With these identifications, we have

Γ/Γ(2) ≅ SL(2, ℤ_{⊭}) ≅ PSL(⊭, ℤ_{⊭})

and

SL(2, ℤ)/Γ(ℸ) ≅ SL(⊭, ℤ_{ℸ}), Γ/Γ(ℸ) ≅ PSL(⊭, ℤ_{ℸ}), ℸ > ⊭.

The orbifold ℍ^{2}/Γ has signature (0, 3; 2, 3, ∞); while for k ≥ 2, Γ(k) is torsion free, and the Riemann surface ℍ^{2}/Γ(k) is of type (p(k), n(k)), where

p(2) = 0

and

p(k) = 1 + \frac{k^{2}(k − 6)}{24} ∏_{𝔓 prime, 𝔓|k} (1 − \frac{1}{𝔓^{2}} ) for k > 2,

while

n(2) = 3

and

n(k) = \frac{k^{2}}{2} ∏_{𝔓 prime, 𝔓|k} (1 − \frac{1}{𝔓^{2}} ) for k > 2.  (1)

If k is an odd prime, then the above formulae simplify to

p(k) = 1 + \frac{(k − 6)(k^{2} − 1)}{24} and n(k) = \frac{k^{2} − 1}{2}.

For k ≥ 2, we have

(k − 6)n(k) = 12(p(k) − 1),  (2)

and

μ(k) = μ(Γ(k)) = 6(2p(k) − 2 + n(k)) = kn(k) = \frac{12k}{k − 6}(p(k) − 1).

For many applications we need to know generators for the cyclic groups stabilizing the parabolic fixed points of Γ(k). It is obvious that B^{k} is a generator for the stabilizer of ∞. The fact that Γ(k) is normal in Γ allows us to compute the stabilizer of an arbitrary x ∈ ℚ ∪ {∞}.

Choose any C ∈ Γ with C(x) = ∞, then C^{−1} ∘ B^{k} ∘ C is a generator of the stabilizer of x in C^{−1}Γ(k)C = Γ(k).

We tabulate, for the convenience of the reader, the type of ℍ^{2}/Γ(k) and μ(k) for low values of k in the following table:

\begin{matrix}\underline{k} & \underline{p(k)} & \underline{n(k)} & \underline{μ(k)} \\ 2 & 0 & 3 & 6 \\ 3 & 0 & 4 & 12 \\ 4 & 0 & 6 & 24 \\ 5 & 0 & 12 & 60 \\ 6 & 1 & 12 & 72 \\ 7 & 3 & 24 & 168 \\ 8 & 5 & 24 & 192 \\ 9 & 10 & 36 & 324 \\ 10 & 13 & 36 & 360 \\ 11 & 26 & 60 & 660 \\ 12 & 25 & 48 & 576 \\ 13 & 50 & 84 & 1092 \end{matrix}

For later use, we define^{2}

n(1) = 1.

Let k = 𝔓^{α}k_{1}, with α and k_{1} positive integers, 𝔓 a positive prime, and 𝔓 |̸k_{1}. Then

n(k) = n(𝔓^{α}k_{1}) = 𝔓^{2α} (1 − \frac{1}{𝔓^{2}} ) n(k_{1}) = 2n(𝔓^{α})n(k_{1}) if k_{1} ≠ 2,

and

n(k) = n(2𝔓^{α}) = \frac{3}{2}𝔓^{2α} (1 − \frac{1}{𝔓^{2}} ) = n(𝔓^{α})n(2) if k_{1} = 2.

In the last formula we have of course assumed that 𝔓 ≠ 2.

2.3. Adjoining translations: the subgroups G(k). Let G(k) be the group generated by the elements of Γ(k) and the translation B. Our first claim is that (see also Lemma 2)

G(k) = {g ∈ Γ; g^{−1}(∞) = ∞ mod Γ(k)}.

It is clear that G(k) is contained in the group defined by the right hand side of the last displayed equation. Conversely, if for g ∈ Γ there is a γ ∈ Γ(k) with g^{−1}(∞) = γ^{−1}(∞), then g ∘ γ^{−1} belongs to Γ and fixes ∞; hence it is a power of B. We have also shown that every element g ∈ G(k) can be decomposed uniquely as

2Thus for all k ∈ ℤ^{+}, n(k) = ν_{∞}(Γ(k)).

g = B^{l} ∘ γ,  (3)

with γ ∈ Γ(k) and l ∈ {0, 1, ..., k − 1}.

2.4. The Hecke subgroups Γ_{o}(k). For each positive integer k, we define the Hecke group Γ_{o}(k) to consist of those Möbius transformations

that can be represented by elements γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ SL(2, ℤ) with

c ≡ 0 mod k.

The branch schema for Γ_{o}(k) is described by [?, Proposition 1.43]. We record for low values of k, this information and the genus p of ℍ^{2}/Γ_{o}(k) in the following table:

\begin{matrix}\underline{k} & \underline{μ} & \underline{ν_{2}} & \underline{ν_{3}} & \underline{ν_{∞}} & \underline{p} \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 2 & 3 & 1 & 0 & 2 & 0 \\ 3 & 4 & 0 & 1 & 2 & 0 \\ 4 & 6 & 0 & 0 & 3 & 0 \\ 5 & 6 & 2 & 0 & 2 & 0 \\ 6 & 12 & 0 & 0 & 4 & 0 \\ 7 & 8 & 0 & 2 & 2 & 0 \\ 8 & 12 & 0 & 0 & 4 & 0 \\ 9 & 12 & 0 & 0 & 4 & 0 \\ 10 & 18 & 2 & 0 & 4 & 0 \\ 11 & 12 & 0 & 0 & 2 & 1 \\ 12 & 24 & 0 & 0 & 6 & 0 \\ 13 & 14 & 2 & 2 & 2 & 0 \end{matrix}

For k an odd prime, the branch schema for Γ_{o}(k) is quite simple. We have

μ = k + 1 and ν_{∞} = 2,
ν_{2} = 2 if k ≡ 1 mod 4 and ν_{2} = 0 if k ≡ 3 mod 4,

and

ν_{3} = 1 if k = 3, ν_{3} = 2 if k ≡ 1 mod 3, and ν_{3} = 0 if k ≡ 2 mod 3.

The cusps ∞ and 0 determine the two punctures on ℍ^{2}/Γ_{o}(k). The

first cusp is stabilized by < B >; the second by [ \begin{matrix}1 & 0 \\ k & 1 \end{matrix} ]. It is of

interest to speculate about the connections, if any, between the branch schema for Γ_{o}(k) and the set of characteristic classes fixed by Γ_{o}(k) (see Lemma 16).

It is obvious that G(k) ⊂ Γ_{o}(k). It follows that G(k) = Γ_{o}(k) for k ≤ 4 and k = 6 because both groups have the same index in Γ.

3. The action of Γ on characteristic classes

3.1. The space of characteristic classes. In [?, §3] we defined a fundamental domain P for the action of Γ on ℝ^{2} ≅ ℂ (interpreted as the set of characteristics) that will be useful (see, for example, Remark 2) to define invariants for characteristic classes. In this regard, we observe that the map

z ↦ f(z) = \frac{θ [ \begin{matrix}1 \\ 0 \end{matrix} ] (z, 2ı)}{θ [ \begin{matrix}0 \\ 0 \end{matrix} ] (z, 2ı)}

from ℂ to ℂ̂ satisfies

f(z + 2) = f(z), f(z + 2ı) = f(z), f(−z) = f(z), all z ∈ ℂ,

has a simple pole at z = \frac{1}{2} + ı, and is regular elsewhere in P. Hence it defines a bijection between the set of characteristic classes^{3}, viewed as the quotient of ℂ by the Euclidean group generated by z ↦ z + 2, z ↦ z + 2ı, and z ↦ −z. The above observation is related to the fact that theta functions with integral characteristics can be used (see, for example, [?]) to develop elliptic function theory; it is also a hint to the claim that theta functions can be used (as shown in §9) to solve a number of conformal mapping problems.

3.2. Rational characteristic classes. A characteristic [ \begin{matrix}x \\ y \end{matrix} ] ∈ ℝ^{2}

is called rational if both x and y are. For the positive integer k, we let Z(k) denote the equivalence classes of characteristics represented

by vectors in ℝ^{2} of the form [ \frac{m}{k} ; \frac{m′}{k} ] with m and m′ in ℤ. (All our

definitions are in terms of characteristics, but apply to characteristic classes. We will normally leave it to the reader to verify this.)

.5The characteristic [ \frac{m}{k} ; \frac{m′}{k} ] ∈ Z(k) is said to be odd or of odd parity

if mm′ is an odd integer, to be even if both m and m′ are even integers, and to be of mixed parity if precisely one of m and m′ is even. This

f(z + 1) = −f(z), z ∈ ℂ.

3The function f does not project to a single valued function on the torus ℂ modulo the group generated by the translations z ↦ z + 1 and z ↦ z + 2ı since