Preprint (June 19, 1999), Version 0.1

Congruences between modular forms

W.A. Stein

Department of Mathematics, University of California, Berkeley, CA 94720, USA

Introduction

This is an account of the paper [Sturm] which is of importance to the explicit computation of Hecke algebras. Suppose f = ∑ a_{n}q^{n} and g = ∑ b_{n}q^{n} are modular forms in S_{k}(Γ). Suppose that the field K generated by all of the a_{n} and b_{n} is a finite extension of Q. Let λ be a prime of K. Sturm deduces an integer B, depending in a simple way on k and the index of Γ in SL_{2}(Z), such that if a_{n} − b_{n} ∈ λ for n ≤ B then a_{n} − b_{n} ∈ λ for all n.

1 Notation and terminology

Fix a positive integer N and let

Γ(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≡ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) mod N\}.

A subgroup Γ of SL_{2}(Z) is a called a congruence subgroup of level N if it contains Γ(N) for some N. Fix such a subgroup Γ. Let

𝔥 = {z ∈ C : Im(z) > 0}

be the complex upper half-plane. For any function f : 𝔥 → C, matrix γ ∈ SL_{2}(Z), and integer k, define f|[γ]_{k} : 𝔥 → C by

(f|[γ]_{k})(z) = (cz + d)^{−k}f (\frac{az + b}{cz + d}) .

Let M_{k}(Γ) be the space of modular forms for Γ of weight k, that is, the complex vector space of function f : 𝔥 → C such that (1) f|[γ]_{k} = f for all γ ∈ Γ, (2) f is holomorphic, and (3) f|[γ]_{k} is bounded on {z : Im(z) > 1} for all γ ∈ SL_{2}(Z). If f ∈ M_{k}(Γ) then f(z) has a Fourier expansion

f(z) = ∑_{\begin{matrix}n≥0 \\ n∈ \frac{1}{N} Z \end{matrix}} a_{n}q^{n}

where q = e^{2πiz} and a_{n} = a_{n}(f) ∈ C. Let S_{k}(Γ) be the subspace of cusp forms, that is, those f ∈ M_{k}(Γ) such that a_{0}(f|[γ]_{k}) = 0 for all γ ∈ SL_{2}(Z). If R is a subring of C, denote by M_{k}(Γ, R) the subset of M_{k}(Γ) consisting of modular forms whose q-expansion coefficients a_{n}(f) lie in R.

2 Congruences

Let k ≥ 1 be an integer, Γ a congruence subgroup, and let μ = [SL_{2}(Z) : Γ]. Let K be a number field (considered as a subfield of C) with ring of integers O. Fix a prime ideal 𝔭 of O and let F = O/𝔭 denote the residue class field. Let O_{𝔭} ⊂ K denote the localization of O at 𝔭, i.e., the domain obtained by inverting all elements not in 𝔭. The valuation on F[[q]] induces a valuation v_{𝔭} on M_{k}(Γ, O_{𝔭}) via the natural map M_{k}(Γ, O_{𝔭}) → F[[q]]. Thus

v_{𝔭}(f) = min{n : a_{n}(f) ≢ 0 (mod 𝔭)}.

Theorem 2.1. If f ∈ M_{k}(Γ, O_{𝔭}) and v_{𝔭}(f) > \frac{μk}{12} , then f ≡ 0 mod 𝔭.

We recall some notation and a lemma before giving the proof. Let

Δ(q) = q − 24q^{2} + 252q^{3} − 1472q^{4} + 4830q^{5} + ⋯ ∈ S_{12}(SL_{2}(Z))
j(q) = \frac{1}{q} + 744 + 196884q + 21493760q^{2} + 864299970q^{3} + ⋯ ∈ K(X(1))

Note that j(q) is an automorphic function for SL_{2}(Z) of weight 0 whose only pole is at infinity (i.e., j|[γ]_{0} = j for all γ ∈ SL_{2}(Z)). The Fourier coefficients of both Δ and j lie in Z.

For any positive integer n, let ζ_{n} = e^{2πi/n} be a primitive nth root of unity.

Lemma 2.2. Let N be a positive integer. Then any γ ∈ SL_{2}(Z) induces a linear isomorphism

[γ]_{k} : M_{k}(Γ(N)) → M_{k}(Γ(N))

which preserves M_{k}(Γ(N), Q(ζ)).

Proof. That [γ]_{k} leaves M_{k}(Γ(N)) invariant follows because Γ(N) is a normal subgroup of SL_{2}(Z): if α ∈ Γ(N) then there is α′ ∈ Γ(N) such that γα = α′γ, so for any f ∈ M_{k}(Γ(N)),

(f|[γ]_{k})|[α]_{k} = f|[γα]_{k} = f|[α′γ]_{k} = (f|[α′]_{k})|[γ]_{k} = f|[γ]_{k}.

Furthermore, [γ]_{k} is an isomorphism because [γ^{−1}]_{k} is its inverse. The second part follows because X(N) has a model over Q(ζ_{N}) and SL_{2}(Z/NZ) acts via automorphisms of the function field of X(N). The action of [γ]_{k} is induced by this action. This is discussed in Chapter 6 of [Shimura].

Proof of Theorem 2.1. We assume that f ≠ 0, otherwise we are done. In each case we will give the argument for O and then note that it works for O_{𝔭} as well.

Case one: Γ = SL_{2}(Z). In the ring O{{q}} of Laurant series,

\frac{f^{12}}{Δ^{k}} = f^{12} · (\frac{1}{q} + 24 + 324q + ⋯)^{k} = ∑_{n≥−k} b_{n}q^{n} ∈ O\{\{q\}\}.

Since

v_{𝔭} (\frac{f^{12}}{Δ^{k}} ) = 12v_{𝔭}(f) − kv_{𝔭}(Δ) = 12v_{𝔭}(f) − k > 0

we see that b_{n} ∈ 𝔭 for n ≤ 0. By inductively subtracting off multiples of powers of j(q) = \frac{1}{q} + 744 + ⋯ from \frac{f^{12}}{Δ^{k}} (reducing the order of the pole by at least one at each step) we find a polynomial R(j) in O[j] such that

\frac{f^{12}}{Δ^{k}} − R(j) ∈ qO[[q]].

But this difference is a holomorphic function on the projective plane, so it must be a constant, hence 0, so that f^{12} = R(j)Δ^{k}. Each of the coefficients of R(j) must lie in 𝔭, otherwise v_{𝔭}(R(j)) ≤ 0 which would be a contradiction because then v_{𝔭}(\frac{f^{12}}{Δ^{k}} − R(j)) ≤ 0. Thus f^{12} = R(j)Δ^{k} ≡ 0 (mod 𝔭) and the same holds for f since (O/𝔭)[[q]] is a domain.

Note that the same argument works with O replaced by its localization at 𝔭. Thus we can allow denominators in the q-expansion of f as long as they do not lie in 𝔭.

Case two: Γ arbitrary. For any γ ∈ SL_{2}(Z), Lemma 2.2 implies that f|[γ]_{k} ∈ M_{k}(Γ(N), K(ζ_{n})). Let 𝔓|𝔭 be a prime ideal of the ring of integers of K(ζ_{n}). Let π be a uniformizer for 𝔓. For any γ, there is a power π^{n} of π so that the coefficients of the q-expansion of π^{n}f|[γ]_{k} are 𝔓-integral and π^{n}f|[γ]_{k} ≢ 0 (mod 𝔓). To do this choose n minimally (possibly negative!) so that π^{n}f|[γ]_{k} is 𝔓-integral. If π^{n}f|[γ]_{k} ≡ 0 (mod 𝔓) then all coefficients are divisible by π so n was not chosen minimally.

Write

SL_{2}(Z) = ⋃_{i=1}^{μ} Γγ_{i}

with γ_{1} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ), and let the n_{i} be chosen for γ_{i} as in the previous paragraph. Then

F = f · ∏_{i=2}^{μ} π^{n_{i}}f|[γ_{i}]_{k} ∈ M_{kμ}(SL_{2}(Z))

and

v_{𝔓}(F) ≥ v_{𝔓}(f) = v_{𝔭}(f) > \frac{μ}{12}k.

By case 1, F ≡ 0 (mod 𝔓), so since π^{n_{i}}f|[γ_{i}]_{k} ≢ 0 (mod 𝔓) for each i ≥ 2, it follows that f ≡ 0 (mod 𝔭) as claimed.

Remark 2.3. For cusp forms the proof of Theorem 1 yields the following slightly improved bound: If f ∈ S_{k}(Γ, O_{𝔭}) and

v_{𝔭}(f) > \frac{μ}{12}k − \frac{μ − 1}{N}

then f ≡ 0 (mod 𝔭).

Proof. Consider the proof of case two. Note that the q-expansion of an element of M_{k}(Γ(N)) is in powers of q^{1/N}. Since f is a cusp form, we know that v_{𝔓}(f|[γ_{i}]_{k}) ≥ \frac{1}{N} for each i. If we insure that

v_{𝔭}(f) > \frac{μ}{12}k − \frac{μ − 1}{N}

then

v_{𝔓}(F) ≥ v_{𝔓}(f) + \frac{μ − 1}{N} = v_{𝔭}(f) + \frac{μ − 1}{N} > \frac{μ}{12}k

and the proof goes through.

3 Applications

Let Γ be a congruence subgroup of level N and index μ, let k be a positive integer, and set r := \frac{k}{12}μ.

Proposition 3.1. Suppose

f = ∑_{\begin{matrix}n>r \\ n∈ \frac{1}{N} Z \end{matrix}} a_{n}q^{n} ∈ M_{k}(Γ)

then f = 0. Stated differently, an element of M_{k}(Γ) is determined by its Fourier coefficients a_{n} for n ≤ r.

Proof. Let s = q^{1/N}. We must show that the composite map

M_{k}(Γ) ↪ C[[s]] → C[[s]]/(sq^{r})

is injective. Let K be a number field with ring of integers O such that M_{k}(Γ) = M_{k}(Γ, O) ⊗ C. Because C is flat over O, it suffices to show that the map Φ : M_{k}(N; O) → O[[s]]/(sq^{r}) is injective. (We are just using that a linear map which is injective over O induces an injective map over C.) Suppose g = ∑ b_{n}q^{n} ∈ M_{k}(N; O) and Φ(g) = 0 so that b_{n} = 0 for n ≤ r. Thus if 𝔭 be a prime of O, b_{n} ≡ 0 (mod 𝔭) for any n ≤ r. By Theorem 2.1, it follows that g ≡ 0 (mod 𝔭). Repeating this argument for all 𝔭 of O shows that the coefficients of g are divisible by all primes 𝔭, i.e., they are 0.

Now suppose Γ = Γ_{0}(N) or Γ_{1}(N) so that we have a Hecke algebra T which acts as a commutative ring of linear endomorphism on M_{k}(Γ).

Theorem 3.2. The Hecke algebra is generated as a Z-module by T_{1}, …, T_{r}.

For the proof, see [Stein].

References

[Shimura] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1994

[Stein] W. Stein, Generating the Hecke algebra as a Z-module

[Sturm] J. Sturm, On the Congruence of Modular Forms. Number theory (New York, 1984–1985), 275–280, Lecture Notes in Math., 1240, Springer, Berlin-New York, 1987.