Generating the Hecke algebra as a Z-module

W. Stein^{∗}

November 28, 1998

Abstract We apply a theorem of Sturm on congruences between modular forms to give a bound for the number of Hecke operators T_{n} needed to additively generate the Hecke algebra associate to weight k modular forms for X_{0}(N).

1 Introduction

In this note we apply a theorem of Sturm [S] on congruences between modular forms to give a bound (Theorem 3.3) for the number of Hecke operators needed to generate the Hecke algebra as a Z-module. The observation that this follows from [S] might be due to Ribet. The case of weight 2 and prime level is treated in [A]. We also use Sturm’s result to prove a “well-known” bound on the number of coefficient needed to determine a modular form (Proposition 3.2).

In section 2 we record our notation and some standard theorems. In section 3 we state Sturm’s theorem and give the above mentioned applications.

2 Modular forms and Hecke operators

Let N and k be positive integers and let M_{k}(N) = M_{k}(Γ_{0}(N)) be the C-vector space of weight k modular forms on X_{0}(N). This space can be viewed as the set of functions f(z), holomorphic on the upper half-plane, such that

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

for all γ ∈ Γ_{0}(N), and such that f satisfies a certain holomorphic condition at the cusps.

Any f ∈ M_{k}(N) has a Fourier expansion

f = a_{0}(f) + a_{1}(f)q + a_{2}(f)q^{2} + ⋯ = ∑ a_{n}q^{n} ∈ C[[q]]

where q = e^{2πiz}. The map sending f to its q-expansion is an injective map M_{k}(N) ↪ C[[q]] called the q-expansion map. Define M_{k}(N; Z) to be the inverse image of Z[[q]] under this map. It is known (see §12.3, [DI]) that

M_{k}(N) = M_{k}(N; Z) ⊗ C.

∗UC Berkeley, Department of Mathematics, Berkeley, CA 94720, USA.

For any ring R, define M_{k}(N; R) := M_{k}(N; Z) ⊗_{Z} R.

Let p be a prime. Define two operators on C[[q]]:

V_{p}(∑ a_{n}q^{n}) = ∑ a_{n}q^{np}

and

U_{p}(∑ a_{n}q^{n}) = ∑ a_{np}q^{n}.

The Hecke operator T_{p} acts on q-expansions by

T_{p} = U_{p} + ε(p)p^{k−1}V_{p}

where ε(p) = 1, unless p|N in which case ε(p) = 0. If m and n are coprime, the Hecke operators satisfy T_{nm} = T_{n}T_{m} = T_{m}T_{n}. If p is a prime and r ≥ 2,

T_{p^{r}} = T_{p^{r−1}}T_{p} − ε(p)p^{k−1}T_{p^{r−2}}.

The T_{n} are linear maps which preserves M_{k}(N; Z). The Hecke algebra T = T(N) = Z[T_{1}, T_{2}, T_{3}, …], which is viewed as a subring of the ring of linear endomorphisms of M_{k}(N), is a finite commutative Z-algebra.

Proposition 2.1. Let ∑ a_{n}q^{n} be the q-expansion of f ∈ M_{k}(N) and let ∑ b_{n}q^{n} be the q-expansion of T_{m}f. Then the coefficients b_{n} are given by

b_{n} = ∑_{d|(m,n)} ε(d)d^{k−1}a_{mn/d^{2}}.

Note in particular that a_{1}(T_{m}f) = a_{m}(f).

Proof. Proposition 3.4.3, [DI].

Proposition 2.2. For any ring R, there is a perfect pairing

T_{R} ⊗_{R} M_{k}(N; R) → R, (T, f) ↦ a_{1}(T f),

where T_{R} = T ⊗_{Z} R.

Proof. Proposition 12.4.13, [DI].

3 Bounding the number of generators

Let μ(N) = N ∏_{p|N}(1 + \frac{1}{p}) be the index of Γ_{0}(N) in SL_{2}(Z).

Theorem 3.1. Let λ be a prime ideal in the ring of integers O of some number field. Suppose f ∈ M_{k}(N; O) is such that a_{n}(f) ≡ 0 (mod λ) for n ≤ \frac{k}{12}μ(N). Then f ≡ 0 (mod λ).

Proof. Theorem 1, [S].

Denote by [x] the largest integer ≤ x.

Proposition 3.2. Suppose f ∈ M_{k}(N) and

a_{n}(f) = 0 for n ≤ r = [\frac{k}{12}μ(N)] .

Then f = 0.

Proof. We must show that the composite map

M_{k}(N) ↪ C[[q]] → C[[q]]/(q^{r+1})

is injective. Because C is a flat Z-module, it suffices to show that the map Φ : M_{k}(N; Z) → Z[[q]]/(q^{r+1}) is injective. Suppose Φ(f) = 0, and let p be a prime number. Then a_{n}(f) = 0 for n ≤ r, hence plainly a_{n}(f) ≡ 0 (mod p) for any such n. By Theorem 3.1, it follows that f ≡ 0 (mod p). Repeating this argument shows that the coefficients of f are divisible by all primes p, i.e., they are 0.

Theorem 3.3. The Hecke algebra is generated as a Z-module by T_{1}, …, T_{r} where r = [\frac{k}{12}μ(N)].

Proof. Let A be the submodule of T generated by T_{1}, T_{2}, …, T_{r}. Consider the exact sequence of additive abelian groups

0 → A ⟶^{i} T → T/A → 0.

Let p be a prime and tensor with F_{p} to obtain

A ⊗ F_{p} ⟶^{\overline{i}} T ⊗ F_{p} → (T/A) ⊗ F_{p} → 0

(tensor product is right exact). Put R = F_{p} in Proposition 2.2, and suppose f ∈ M_{k}(N, F_{p}) pairs to 0 with each of T_{1}, …, T_{r}. Then by Proposition 2.1, a_{m}(f) = a_{1}(T_{m}f) = 0 in F_{p} for each m, 1 ≤ m ≤ r. By Theorem 3.1 it follows that f = 0. Thus the pairing, when restricted to the image of A ⊗ F_{p} in T ⊗ F_{p}, is also perfect and so

dim_{F_{p}} \overline{i}(A ⊗ F_{p}) = dim_{F_{p}} M_{k}(N, F_{p}) = dim_{F_{p}} T ⊗ F_{p}.

We see that (T/A)⊗F_{p} = 0; repeating the argument for all p shows that the finitely generated abelian group T/A must be trivial.

References

[A] A. Agashe, On the Calculation of Eisenstein quotient. Preprint, 1998.

[DI] F. Diamond, J. Im, Modular forms and modular curves. Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994), 39–133, CMS Conf. Proc., 17, Amer. Math. Soc., Providence, RI, 1995.

[L] S. Lang, Introduction to modular forms. Grundlehren der Mathematischen Wissenschaften, 222. Springer-Verlag, Berlin, 1995.

[S] 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.