On Modular Functions with a Cuspidal Divisor

ON MODULAR FUNCTIONS WITH A CUSPIDAL DIVISOR Quentin Gazda∗

arXiv:1609.03872v1 [math.NT] 13 Sep 2016

École Normale Supérieure September 14, 2016

Abstract The aim of this paper is the generalization of the following equivalence due to Kohnen (in [6]): Let f be a modular function of integral weight with respect to Γ0 (N ), N square-free. Then f has a cuspidal divisor (i.e. zeros and poles supported at the cusps) if and only if f is an η-quotient. We first present Theorem 3, which states that Kohnen’s result still holds when N = 4M and 8M , with M an odd and square-free integer, and then Theorem 4, which extends the equivalence to the cases • N = 9M and N = 27M , with M a square-free integer coprime to 3, • N = 16M and N = 32M , with M an odd and square-free integer, • N = 25M and N = 125M , with M a square-free integer coprime to 5, by introducing a generalization of the classical Dedekind η-function.

I

Introduction to the main result.

Let f be a meromorphic modular function of integral weight k on a congruence subgroup Γ of finite index in Γ(1) := SL2 (Z), i.e. it is a meromorphic function on the complex upper half-plane H, satisfies the usual transformation in weight k under the action of Γ, and is meromorphic at the cusps. In the present paper we shall try to characterize f when it has no zeros or poles on H. In the context of Riemann surfaces, we usually say that the function has a cuspidal divisor (in the sense that zeros and poles are supported at the cusps). When Γ = Γ(1), one concludes easily from the valence formula that 12|k and f is proportional to ∆k/12 where ∆ is the modular discriminant. In 2003, Winfried
Kohnen extended this well-known result a b ∈ Γ(1), N |c where N is a square-free integer (see [6], to the case Γ0 (N ) = c d Thm. 2): he showed that f was an η-quotient of level N in the following sense: given f1 , f2 , ..., fd , complex-valued functions from the complex upper half-plane, a function f from H to C is a f1 , f2 , ..., fd -quotient of level N if there exist complex Q numbers c and (at,i )t|N,1≤i≤d such that f = c t|N (f1 |Vt )at,1 · · · (fd |Vt )at,d where we set (fi |Vt )(τ ) = fi (tτ ) (for all τ ∈ H) and where the complex powers are defined by the principal branch of the complex logarithm. As usual, η denotes the Dedekind function ∞ Y η(τ ) = q 1/24 (1 − q n ) (q = e2iπτ ). n=1

In Kohnen’s paper, following Thm. 2, one can read: ∗ [email protected]

1

On Modular Functions with a Cuspidal Divisor

Quentin Gazda

We think that the assertion of Theorem 2 or a slightly weaker statement eventually would be true for arbitrary N , but it seems that this cannot be proved by the method employed here. Surprisingly, this Theorem 2 appears to fail for some smaller congruence subgroups but via a generalization of the Dedekind η function this result can be extended. For χ a Dirichlet character modulo N (N > 1), we will refer to the following function: ηχ (τ ) :=

∞ Y

¯ ¯ ¯ (1 − ζu q n )χ(n) (1 − ζu2 q n )χ(2n) · · · (1 − ζuu q n )χ(un)

(τ ∈ H)

n=1

where ζu := e2iπ/u , as the Generalized Dedekind η function attached to χ. Denoting 1N the principal character modulo N , we convience that η11 is the classical Dedekind function. This convention will make sense in the next section. We shall first explain why they are natural generalizations of η and show that ηχ satisfies a particular modular transformation under the action of a Hecke congruence subgroup. More precisly, we prove: Theorem 1. Let χ be a real Dirichlet character (i.e. Q taking its values in {−1, 1}) with modulus N > 1 and conductor u. Let Q := u2 p|N,p∤u p. Then ηχ is a Parabolic Generalized Modular Function (of weight 0) on Γ0 (Q) with a cuspidal divisor. Further, ηχ is an η-quotient if and only if χ is principal. As it is less well-known, we shall recall the definition of a Generalized Modular Function in the next section. The situation appears to be more complicated when χ is non real. Indeed, when χ takes non real values, the function ηχ is not a Generalized Modular Function anymore. By now, even when χ is real, it is not clear whether ηχ is a classical meromorphic modular function with a cuspidal divisor for all modulus N . Nevertheless we have the following statement: Theorem 2. For an odd prime number p, let (·/p) denotes the Legendre symbol attached to p and let ψ denotes the non principal (real) Dirichlet character mod 4. Then η( · ) , ηψ and η( · ) are Generalized Modular Functions with unitary characters, 3

5

taking values in the twelfth roots of unity, on the respective groups Γ0 (9), Γ0 (16) and Γ0 (25) with a cuspidal divisor and which are not η-quotient. Equivalently, the twelfth powers of η( · ) , ηψ and η( · ) are meromorphic modular 3 5 functions with a cuspidal divisor. Namely, those functions are: ( ) ∞  Y 1 − e2iπ/3 q n 3 n

η( · ) (τ ) = 3

n=1

1 − e4iπ/3 q n

η( · ) (τ ) := 5

∞ Y

n=1

1− 1−

, ηψ (τ ) =

√ 1− 5 n 2√ q 1+ 5 n 2 q

ψ(n) ∞  Y 1 − iq n

n=1

+ q 2n + q 2n

!( n5 )

1 + iq n

,

(τ ∈ H).

After proving those theorems, we will essentially introduce two results. The first one is a extension of Kohnen’s theorem: Theorem 3. Let N be an integer of the form 4M or 8M where M is odd and squarefree. Any meromorphic modular function of integral weight on Γ0 (N ) is an η-quotient of level N . The second one partially answers to Kohnen’s expectations: Theorem 4. Let χ1 and χ2 be the two primitive Dirichlet characters modulo 5 distinct from (·/5). Any meromorphic modular function of integral weight on Γ0 (N ) with a cuspidal divisor can be expressed as

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On Modular Functions with a Cuspidal Divisor

Quentin Gazda

(i) an η, η( · ) -quotient of level N when N is of the form 9M or 27M with M square3 free and coprime with 3, (ii) an η, ηψ -quotient of level N when N is of the form 16M or 32M with M odd and square-free, (iii) an η, η( · ) , ηχ1 , ηχ2 -quotient of level N when N is of the form 25M or 125M 5 with M square-free and coprime with 5. We deeply think that the method employed here can be extended to prove some Thm. 4-like results for other congruence subgroups and hopefully finish the classification for arbitrary N .

II

Early results and notations.

The proofs of those four theorems use extensively the basic ingredients in the theory of Generalized Modular Functions (usually refered as GMFs) introduced by Marvin Knopp and Geoffrey Mason in [5]. Let us briefly recall the story of GMFs. Let Γ ⊂ Γ(1) be a subgroup of finite index. A Generalized Modular Function (of weight 0) on Γ is a holomorphic function f defined on H such that (i) (f |0 γ) = ν(γ)f for all γ ∈ Γ (where |0 denotes the usual slash operator of weight 0) and where ν : Γ → C∗ is a (not necessarily unitary) character of Γ. (ii) f is meromorphic at the cusps of Γ. Further, if f satisfies the following additional condition (iii) ν(γ) = 1 for every element γ ∈ Γ of trace 2. then we call f a parabolic GMF (PGMF) with multiplier system ν (of weight 0). One main result of Knopp and Mason can be stated as follow (see [5], Thm. 2 for a general version): if f is a PGMF on Γ with a cuspidal divisor (i.e. zeros and poles supported at the cusps), then its q-logarithmic derivative g :=

θf 1 f′ = f 2iπ f

(1)

is a classical modular form of weight 2 on Γ with trivial character and statisfies (a) The constant term in the expansion of g at every cusps s = a/c of Γ takes the form ls /(ws c2 ) where ls ∈ Z (namely ords (f )) and ws is the width of s. Conversely, if g is a classical modular form of weight 2 on Γ with trivial character and statisfies condition (a), there is a PGMF f with a cuspidal divisor (uniquely determined up to multiplication with non-zero complex numbers) which satisfies Eq. (1). We easily derive from this that any PGMF f with unitary multiplier system whose q-logarithmic derivative is a cusp form must be constant. Indeed, |f | is bounded on H and therefore constant by elementary complex analysis. There is also a helpful remark made by Knopp and Mason: if a subgroup of finite index Γ is generated by only parabolic and elliptic elements, then a parabolic multiplier system ν (i.e. attached to a PGMF) must have finite order: by definition ν(P ) = 1 for parabolic elements P and since elliptic elements E (of Γ(1)) have finite order dividing 12, ν(E) is a twelfth root of unity. We conclude that ν has finite order dividing 12 from the fact that Γ is finitely generated. This situation happen when Γ is the group Γ0 (9), Γ0 (16) and Γ0 (25) since those groups have genus 0. We also need to agree on some notations (we will try to use the same of those in the famous textbook of Diamond and Shurman [1]). 3

On Modular Functions with a Cuspidal Divisor

Quentin Gazda

• The vector-valued Eisenstein series of weight 2 modulo N (¯ v = (cv , dv ) ∈ (Z/N Z)2 ) ∞ 4π 2 X v¯ ¯ n σ (n)qN Gv2¯ (τ ) := δ(¯ cv )ζ dv (2) − 2 N n=1 1 where

δ(¯ cv ) :=

(

1 0

if cv ≡ 0 (mod N ), , otherwise. X

σ1v¯ (n) :=

¯

X

ζ dv (2) :=

m≡dv (N ) m6=0

|d|e2iπ

dv d N

1 , m2

,

d|n n/d≡cv (N )

where the two sums are taken over positive and negative values of m and d and where the overline denotes the reduction modulo N . • For ψ and ϕ two Dirichlet characters modulo u and v, N = uv, we define Gψ,ϕ := 2

(cv,d+ev)

X

ψ(c)ϕ(d)G ¯ 2

.

0≤c,e 1 and conductor u. Write χ = χ0 1N where χ0 is a primitive Dirichlet character of conductor u. Then, Y (ηχ0 |Vt )χ0 (t)µ(t) ηχ = t|N

where µ denotes the Möbius function. The proof is straightforward, as an application of Möbius transformation formula, and thus left to the reader (remember that χ0 is real so there is no issue with the complex logarithm). From now on we will assume that χ is primitive. To determine whether ηχ is a PGMF, and in the scope of condition (a), we have to find the constant term of θηχ /ηχ at any cusp and apply Knopp and Mason’s theorem. Lemma 3. For an integer u > 1, let χ be a primitive Dirichlet character with conductor u. The constant term of θηχ /ηχ at every cusps of the form k/u, where gcd(k, u) = 1, is given by  u 2 χ(k) L(2, χ2 ). 2π The function θηχ /ηχ vanishes at the other cusps. 5

On Modular Functions with a Cuspidal Divisor

Quentin Gazda

Proof. According to Lemma 1 it is enough to study E2χ,χ¯ at cusps. It turns out to χ ¯ be easier to calculate the constant term of Gχ, and to use the relation −g(χ)E2χ,χ¯ = 2 χ, χ ¯ (u/2π)2 G2 . It is not difficult to see that for any vector v¯ = (cv , dv ) ∈ (Z/u2 Z)2 , the corresponding vector-valued Eisenstein series of weight 2 modulo u2 , namely Gv2¯ , behaves like a quasi-modular form of weight 2 with respect to   
a b Γ(u2 ) = ∈ Γ(1), ac db ≡ I2 (mod u2 ) . c d More precisly, we have the general transformation property   2iπcγ aγ b γ ∈ Γ(1), (Gv2 |2 γ)(τ ) = Gvγ ∀γ = 2 (τ ) − 4 cγ dγ u (cγ τ + dγ )

χ ¯ where |2 denotes the usual slash operator of weight 2. We know that Gχ, is a modular 2 form with respect to Γ0 (u2 ) (remember that neither χ or χ ¯ is principal). We thus have the following transformation property: χ ¯ (Gχ, 2 |2 γ) =

X

(cu,d+eu)γ

χ(c)χ(d)G2

, since

0≤c,d,e