Congruences between Modular Forms

M. Ram Murty*

1. Introduction Recently, Goldfeld and Hoffstein [4] have shown, using the theory of L-functions that if f and g are two holomorphic Hecke newforms of weight k and squarefree levels N_{1}, N_{2} respectively, then there is an n = O(N log N), with N = lcm(N_{1}, N_{2}) so that

a_{f}(n) ≠ a_{g}(n).

(Here, a_{f}(n) denotes the n-th Fourier coefficient in the Fourier expansion of f at i∞.) In other words, there is a constant c so that the first cN log N Fourier coefficients determine the newform. They obtain an analogous result if the weights are distinct. Assuming the generalized Riemann hypothesis for the Rankin-Selberg L-functions attached to these eigenforms, they deduce that the bound above can be improved to O((log N)^{2}(log log N)^{4}).

We will show that these results can be established without the use of Lfunctions. Our approach leads to sharper results and is applicable in the wider context of two arbitrary cusp forms of any weight and level. In fact, we will prove a more general and sharper:

Theorem 1 Let f and g be two distinct holomorphic modular forms of weight k and levels N_{1} and N_{2} respectively. Let N = lcm(N_{1}, N_{2}). Then, for some

n ≤ \frac{k}{12}N ∏_{p|N} (1 + \frac{1}{p})

we must have a_{f}(n) ≠ a_{g}(n). (Here, the product is over primes p dividing N.)

Remark Note that we do not assume that f and g are Hecke eigenforms nor that they are of squarefree levels as in the Goldfeld - Hoffstein [4] paper. Moreover, let us observe that if ν(N) denotes the number of prime factors of N, and p_{i} denotes the i-th prime, then

∏_{p|N} (1 + \frac{1}{p}) ≤ ∏_{1≤i≤ν(N)} (1 + \frac{1}{p_{i}} ) ≤ ∏_{1≤i≤ν(N)} (1 − \frac{1}{p_{i}} )^{−1} ≪ log log N

* Research partially supported by NSERC, FCAR and CICMA.

which is a sharper bound than the one given in [4, p. 387].

Proof Let ϕ = f − g, and suppose a_{f}(n) = a_{g}(n) for n ≤ M. Then, ϕ has a zero of order ≥ M at i∞. If Δ denotes Ramanujan’s cusp form, then ϕ^{12}/Δ^{k} is of weight zero and hence a meromorphic function on the compact Riemann surface X_{0}(N). Since Δ does not vanish on the upper half-plane, and has a simple zero at i∞, the number of zeroes of ϕ, which is at least 12M − k, is equal to the number of poles, which cannot exceed k times the index of Γ_{0}(N) in Γ(1) minus one (to account for i∞ which already contributed to the zero count). The index of Γ_{0}(N) in Γ(1) is equal to

N ∏_{p|N} (1 + \frac{1}{p}) .

The result is now immediate.

We can make a few remarks. The first is that the method can be applied to any discrete subgroup contained in Γ(1) to get an analogous result. One can also adapt it to deal with f and g of different weights.

The proof of Theorem 1 can be modified to handle different weights. Indeed, if

a_{n}(f) = a_{n}(g)

for all n ≤ M, then

ϕ = (f^{k_{2}} − g^{k_{1}})^{12}/Δ^{k_{1}k_{2}}

is a function on X_{0}(N). The order of the zero at i∞ is ≥ 12M − k_{1}k_{2}. The number of poles, on the other hand is ≤ k_{1}k_{2}([Γ(1) : Γ_{0}(N)]−1) by an analogous argument as before.

Let us introduce the following notation. Suppose

f(z) = ∑_{n=1}^{∞} a_{n}(f)e^{2πinz}

is the Fourier expansion of f at i∞. Then, define

ord_{∞}(f) = min{n : a_{n}(f) ≠ 0}.

We have therefore proved:

Theorem 2 If f and g are holomorphic modular forms such that

ord_{∞}(f − g) > \frac{k_{1}k_{2}}{12} ([Γ(1) : Γ_{0}(N)])

then f = g.

The result which we state below will be proved by a different method in Section 5 as an application of the Riemann-Roch theorem and is a small improvement of Theorem 2.

Theorem 3 Suppose that f and g are two holomorphic cusp forms of weights k_{1} and k_{2} and levels N_{1} and N_{2} respectively. If

ord_{∞}(f − g) > k_{1}k_{2}(μ − 1),

where μ is the genus of X_{0}(N) and N = lcm(N_{1}, N_{2}), then f = g.

Let us note that the genus μ satisfies the inequality

μ ≤ 1 + \frac{N}{12} ∏_{p|N} (1 + \frac{1}{p})

and thus, the bound of Theorem 3 is comparable to the one of Theorem 1.

In case that f and g are normalized newforms of distinct weights k_{1} and k_{2} and levels N_{1} and N_{2} respectively, we can in fact do better and derive an estimate superior to the conditional estimate of Goldfeld and Hoffstein [4, p. 386].

Theorem 4 Let f and g be two holomorphic Hecke newforms of distinct weights k_{1} and k_{2} on Γ_{0}(N_{1}) and Γ_{0}(N_{2}) respectively. Then, there is an n < 4(log N)^{2} with N = lcm(N_{1}, N_{2}) so that

a_{n}(f) ≠ a_{n}(g).

Proof We can view f and g as cusp forms on Γ_{0}(N). Let us first note that there is a p < 2 log N which is coprime to N for otherwise N would be divisible by all the primes < 2 log N and hence by their product. By a classical estimate of Chebycheff, this product is

∏_{p<2 log N} p = exp ( ∑_{p<2 log N} log p) > N,

which is a contradiction. Thus, fixing such a prime p, and observing that

a_{p^{2}}(f) = a_{p}^{2}(f) − p^{k_{1}−1}

and

a_{p^{2}}(g) = a_{p}^{2}(g) − p^{k_{2}−1}

we deduce that k_{1} = k_{2}. Noting that p^{2} < 4 log^{2} N, we have a contradiction.

If we view Theorems 1, 2 and 3 as statements at the “infinite prime” then it is natural to ask for analogous results at the “finite primes”. That is, if we have a congruence between coefficients of modular forms (mod p) up to a certain natural number, can we conclude that the coefficients are always congruent? In the case the two forms have equal weight, such a result was first established by Sturm [12]. Sturm’s proof is sketchy in some places and therefore, for the sake of clarity of exposition and emphasis with the analogy above, we give the complete proof in Sections 2, 3 and 4. Sturm’s argument however cannot be easily modified to handle different weights. In Section 5, we therefore take a different approach through the Riemann-Roch theorem. This has the merit of being conceptually simple and at the same time working (mod p) (for p not dividing N = lcm(N_{1}, N_{2}), however) thanks to the algebrogeometric generalization of the Riemann-Roch theorem. Recently, K. Ono [7] applied the theorem of Sturm in investigating the parity of the partition function.

2. Preliminaries In our paper [8], we indicated how the celebrated ABC conjecture leads naturally to the problem of congruences between modular forms. We will not discuss this connection here, but refer the reader to the forthcoming paper [8] for a detailed derivation. The purpose of this paper is to determine the (finite) amount of calculation necessary in order to establish a congruence between two modular forms.

More precisely, let us fix an algebraic number field F with ring of integers O_{F}. Fix a prime ideal 𝔭 of O_{F} and for a formal power series

s = ∑_{n=0}^{∞} c_{s}(n)q^{k} , c_{s}(n) ∈ O_{F} ,

define

ord_{𝔭}(s) = min{n : 𝔭 ∤ c_{s}(n)}

with the convention that ord_{𝔭}(s) = ∞ if 𝔭|c_{s}(n) for all n. Recall that k[[q]] with k = O_{F} /𝔭 is a discrete valuation ring. In particular, this implies the following. Notice that if

f = ∑_{n≥0} c_{f}(n)q^{n}

g = ∑_{n≥0} c_{g}(n)q^{n}

then

c_{fg}(n) = ∑_{i+j=n} c_{f}(i)c_{g}(j),

so that, if ord_{𝔭}(f) = n_{f} < ∞ and ord_{𝔭}(g) = n_{g} < ∞, then from the above formula, we see

\begin{aligned}c_{fg}(n_{f} + n_{g}) & ≡ c_{f}(n_{f})c_{g}(n_{g}) mod 𝔭 \\ & ≢ 0 mod 𝔭. \end{aligned}

We also note that c_{fg}(n) ≡ 0 mod 𝔭 if n < n_{f} + n_{g}. Hence,

ord_{𝔭}(fg) = ord_{𝔭}(f) + ord_{𝔭}(g)

when each of the terms on the right hand side is finite. By the convention made above, the equality also holds if either one of ord_{𝔭}(f) or ord_{𝔭}(g) is infinity.

For each positive integer N, 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 \}

and fix Γ, a subgroup of Γ(1) containing Γ(N). As usual, 𝔥 will denote the upper half-plane, k will be a positive integer and we will consider functions

f : 𝔥 → ℂ

satisfying certain conditions. M_{k}(Γ) will denote the ℂ-vector space of modular forms of weight k for Γ. To be precise, let us define for each γ ∈ GL_{2}^{+}(ℝ), the function

(f|γ)(z) = f(\frac{az + b}{cz + d})(cz + d)^{−k}(ad − bc)^{k/2}

where

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} )

is a matrix of GL_{2}(ℝ) of positive determinant. Then M_{k}(Γ) consists of holomorphic functions of the extended upper half-plane:

f : 𝔥^{∗} = 𝔥 ∪ ℚ ∪ {i∞} → ℂ

satisfying f|γ = f for all γ ∈ Γ.

Since

( \begin{matrix}1 & N \\ 0 & 1 \end{matrix} ) ∈ Γ,

such an f ∈ M_{k}(Γ) has a Fourier expansion of the following type:

f(z) = ∑_{{n≥0 \atop n∈N^{−1}ℤ}} a_{f}(n)e(nz)

where e(x) = e^{2πix}. If R is a subring of ℂ, we denote by M_{k}(Γ, R) those forms of M_{k}(Γ) whose Fourier coefficients lie in R. That is, a_{f}(n) ∈ R for all n.

Our purpose now is to explain in some detail a fundamental result of Sturm [12] regarding congruences between modular forms. To this end, let us fix as before an algebraic number field F and let O_{F} be the ring of integers of F. Let 𝔭 be a prime ideal of O_{F}. We will explain the following:

Theorem 5 (Sturm) Let f, g ∈ M_{k}(Γ, O_{F}). Suppose ord_{𝔭}(f − g) > k[Γ(1) : Γ]/12. Then f ≡ g mod 𝔭 with Γ a congruence subgroup.

Notice that the bound does not depend on 𝔭.

3. Sturm’s theorem: the level one case To prove the theorem, we first consider the level one case. That is, N = 1. Recall that Ramanujan’s cusp form

Δ(z) = ∑_{n=1}^{∞} τ(n)e(nz) ∈ M_{12}(Γ(1), ℤ)

and can be written in terms of the standard Eisenstein series:

Δ(z) = \frac{1}{1728}(E_{4}^{3} − E_{6}^{2})

with

E_{4}(z) = 1 + 240 ∑_{n=1}^{∞} σ_{3}(n)e(nz),

E_{6}(z) = 1 − 504 ∑_{n=1}^{∞} σ_{5}(n)e(nz)

and

σ_{k}(n) = ∑_{{d|n \atop d>0}} d^{k}.

Also, the modular function j(z) is

j(z) = E_{4}^{3}/Δ.

We reproduce below a result that is well-known for M_{k}(Γ(1)). We adapt it to the case M_{k}(Γ(1), O_{F}).

Proposition 6 Let Φ ∈ M_{12k}(Γ(1), O_{F}), satisfying ord_{𝔭}(Φ) > k. Then

Φ/Δ^{k} ∈ 𝔭[j]

is a polynomial in j of degree at most k, all of whose coefficients are divisible by 𝔭.

Remark This fact is stated without proof in [12]. It is more or less evident from the fact that any modular function with only a pole at infinity must be a polynomial in j. However, the divisibility of the coefficients is not so clear. One can see this by comparing q-expansions. For the sake of completeness, we give a proof that assumes minimal background.

Proof We induct on k. For k = 1, Φ has weight 12 and so we can write it as an O_{F} -linear combination of E_{4}^{3} and Δ, as is easily checked. Dividing by Δ gives the result. Since ord_{𝔭}Φ/Δ > 0, we observe that writing

Φ/Δ = ∑_{n≥−1} c(n)e(nz),

we have c(n) ∈ O_{F} and 𝔭|c(n) if n ≤ 0.

For general k, let us find i and j so that

12k = 4i + 6j.

Then for some c ∈ O_{F},

Φ − cE_{4}^{i}E_{6}^{j}

is a cusp form of weight 12k. Thus, we can write

Φ = cE_{4}^{i}E_{6}^{j} + Δf_{1}

with f_{1} ∈ M_{12(k−1)}(Γ(1), O_{F}). Dividing by Δ^{k} yields

Φ/Δ^{k} = cE_{4}^{i}E_{6}^{j}/Δ^{k} + f_{1}/Δ^{k−1}.

By induction hypothesis,

f_{1}/Δ^{k−1} ∈ 𝔭[j].

Noting that 4i + 6j = 12k implies that i ≡ 0 mod 3 and j ≡ 0 mod 2 we can write i = 3i_{0} , j = 2j_{0} so that

E_{4}^{i}E_{6}^{j}/Δ^{k} = (E_{4}^{3}/Δ)^{i_{0}}(E_{6}^{2}/Δ)^{j_{0}}

and E_{4}^{3}/Δ = j , E_{6}^{2}/Δ = j −1728. This completes the proof of the proposition.

We can now prove the theorem in the level one case. Let ϕ = f − g. Then, ord_{𝔭}(ϕ^{12}) > k implies

ϕ^{12}/Δ^{k} = ∑_{n≥−k} c(n)e(nz)

with c(n) ∈ O_{F} and 𝔭|c(n) if n ≤ 0. By the proposition, ϕ^{12}/Δ^{k} ∈ 𝔭[j] is a polynomial in j of degree at most k. Thus, ϕ^{12} ∈ Δ^{k}𝔭[j] implies ord_{𝔭}ϕ^{12} = ∞ so that ord_{𝔭}ϕ = ∞, as desired.

4. Sturm’s theorem: the general level case We begin by discussing some preliminaries.

Let Γ contain Γ(N). We want to reduce the proof of the Theorem to the level one case by constructing a map:

T : M_{12k}(Γ(N), O_{F})→M_{12k}(Γ(1), O_{F})

such that

ord_{𝔭}(T (ϕ)) ≥ ord_{𝔭}(ϕ).

Theorem 3.52 of Shimura [10, p. 85] assures us that the space S_{k}(Γ(N)), the space of cups forms of weight k for Γ(N) has a basis of cusp forms whose Fourier coefficients at i∞ are rational integers, provided k ≥ 2. This means that any element of S_{k}(Γ(N), F) has the bounded denominator property. That is, given an element f ∈ S_{k}(Γ(N), F), there is an element A ∈ F so that Af ∈ S_{k}(Γ(N), O_{F}).

By the theory of Eisenstein series, we conclude that M_{k}(Γ(N)) has a basis whose Fourier coefficients are rational over ℚ(ζ_{N}) where ζ_{N} denotes a primitive N-th root of unity. (See also Theorems 6.6 and 6.9 of Shimura [10, pp. 136-140]).

Now we can prove the theorem in the general case. As in [12, p. 276], let ϕ = f − g. Our aim is to show that under the hypotheses of the Theorem, ord_{𝔭}(ϕ) = ∞. Thus, replacing ϕ by ϕ^{12} if necessary, we may suppose 12|k. Then, ϕΔ^{−k/12} is a modular function of level N. Since Γ(N) is a normal subgroup of Γ(1), we note that for any γ ∈ Γ(1),

ϕ|γ ∈ M_{k}(Γ(N) , F (ζ_{N})).

By what we have said in the previous paragraph, ϕ|γ has bounded denominators. Now let K be the Hilbert class field of F (ζ_{N}). Then 𝔭O_{K} is a principal ideal in O_{K}. Let ℘ be a prime ideal of O_{K} dividing 𝔭O_{K}.

For every γ ∈ Γ(1), we can clearly find A(γ) ∈ K^{∗} such that ord_{℘}A(γ)(ϕ|γ) is finite, simply by dividing by a suitable power of ℘. Moreover, by the Chinese remainder theorem, we can arrange

A(γ)(ϕ|γ) ∈ M_{k}(Γ(N) , O_{K})

Now consider the “norm function from Γ(N) to Γ(1)”, namely,

Φ = ϕ ∏_{i=2}^{m} A(γ_{i})(ϕ|γ_{i})

where 1 = γ_{1} , γ_{2} , ..., γ_{m} is a set of coset representatives of Γ(N) in Γ(1) Note that

Φ ∈ M_{km}(Γ(1))

because if σ ∈ Γ(1),

Φ|σ = (∏_{i=2}^{m} A(γ_{i})) ∏_{i=1}^{m}(ϕ|γ_{i}σ)

and γ_{1}σ , γ_{2}σ , ... γ_{m}σ is again a set of coset representatives of Γ(N) in Γ(1) so that Φ|σ = Φ. Note also that

ord_{℘}(Φ) ≥ ord_{℘}ϕ = ord_{𝔭}ϕ > km/12.

By the level one cases we deduce that ord_{℘}Φ = ∞. Thus,

ord_{℘}ϕ + ∑_{i=2}^{m} ord_{℘}(A(γ_{i})ϕ|γ_{i}) = ∞

Since ord_{℘}(A(γ_{i})ϕ|γ_{i}) < ∞ for i = 2, ..., m, we conclude that ord_{℘}ϕ = ∞. Hence ord_{𝔭}ϕ = ∞. This completes the proof.

5. An application of the Riemann-Roch theorem

Theorem 7 Suppose that f and g are holomorphic cusp forms on Γ_{0}(N), of weight k and levels N_{1}, N_{2} respectively. Let N be the lcm of N_{1} and N_{2}. Suppose that

ord_{∞}(f − g) > \frac{k}{2} (2μ − 1)

where μ is the genus of X_{0}(N). Then f = g.

Proof If f ≠ g, then k is even since there are no non-zero odd weight forms. ω = (f − g)(dz)^{k/2} is a holomorphic differential k/2-form on X_{0}(N). Its degree is (k/2)(2μ − 2). On the other hand, the hypothesis means that at i∞, ord_{i∞}(ω) ≥ (k/2)(2μ − 1) − (k/2) where the extra k/2 comes from (dz)^{k/2}. Thus, (see for example, Shimura [10, Prop. 2.16, p. 39])

\frac{k}{2} (2μ − 2) = deg(ω) ≥ ord_{i∞}(ω) ≥ \frac{k}{2} (2μ − 1) − \frac{k}{2} ,

which is a contradiction.

Proof of Theorem 3 It is now clear that Theorem 3 can be proved in an exactly similar manner. Indeed, as before, let us consider the differential (f^{k_{2}} − g^{k_{1}})(dz)^{k_{1}k_{2}/2} and proceed as in the previous proof.

Theorem 8 Suppose f and g are cusp forms of even weight k and level N with coefficients lying in the ring of integers O_{F} of some algebraic number field F. Suppose that for some prime ideal 𝔭 coprime to the level N, we have

ord_{𝔭}(f − g) > \frac{k}{2} (2μ − 1),

then, f ≡ g mod 𝔭.

Proof We apply the Riemann-Roch theorem valid in any field of characteristic p which is coprime to N. This uses the non-trivial fact that X_{0}(N) has good reduction for all p ∤ N and is due to Igusa [5] (see also Deligne–Rapoport [3]). Again, cusp forms of weight 2 can be interpreted as differentials on X_{0}(N) over 𝔽_{p}. The same argument as before is valid for arbitrary even weight k. (See Silverman [11, p. 39] and [2, p. 96], for example.)

This approach has the advantage that it can generalize to two different weights k_{1} and k_{2}.

Theorem 9 Suppose that f and g are cusp forms of weights k_{1} and k_{2} and levels N_{1} and N_{2} respectively. Suppose further that at least one of k_{1} or k_{2} is even. As before, let us suppose the coefficients lie in the ring of integers O_{F} of some algebraic number field. If 𝔭 is a prime ideal of O_{F}, and

ord_{𝔭}(f − g) > k_{1}k_{2}(μ − 1)

then f ≡ g mod 𝔭.

We can derive better variations of Theorems 8 and 9 which are better for small primes, if we are willing to assume the generalized Riemann hypothesis for certain Dedekind zeta functions. In fact, if f and g are normalized Hecke eigenforms of level N, we know by Deligne [1] that there exists a Galois extension K_{f}/ℚ and a representation

ρ_{𝔭,f} : Gal(K_{f}/ℚ)→GL_{2}(O_{K_{f}} /𝔭)

such that for each prime v ∤ 𝔭N, we have

tr(ρ_{𝔭,f}(σ_{v}(K_{f}/ℚ))) ≡ a_{v}(f) mod 𝔭

where σ_{v} denotes the Artin symbol of v. An identical result holds for g. Thus, we may consider the compositum K_{f}K_{g} and deduce by the Chebotarev density theorem (see [6] or [9]) the following:

Theorem 10 Assume that the Dedekind zeta function of K_{f}K_{g}/ℚ satisfies the analogue of the Riemann hypothesis and f and g are normalized Hecke eigenforms as above with Fourier coefficients lying in a field F. If 𝔭 is a prime ideal of O_{F} coprime to N and

ord_{𝔭}(f − g) > (log(Norm_{F/ℚ}(𝔭)) + log N)^{4},

then f ≡ g mod 𝔭.

Proof By the Chebotarev density theorem and the Riemann hypothesis for the Dedekind zeta function of K_{f}K_{g}/ℚ, we deduce that for any given conjugacy class C of Gal(K_{f}K_{g}/ℚ), there is a prime v with

Norm(v) ≤ (log(Norm_{F/ℚ}(𝔭)) + log N)^{4}

so that σ_{v}(K_{f}K_{g}/ℚ) ∈ C. Thus, if

a_{v}(f) ≡ a_{v}(g) mod 𝔭

for each v whose norm satisfies the last inequality, then by Deligne’s theorem and the effective Chebotarev density theorem as cited above, we can conclude the desired result.

We can also remark that even if f and g are not Hecke eigenforms, a similar result can still be established. Note however, these bounds depend on 𝔭. Also worthy of contrast is that Theorem 5 is valid for all primes 𝔭 whereas Theorems 8 and 9 are applicable only when 𝔭 is coprime to N.

References

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[9] J.-P. Serre. Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Etudes Sci. Publ. Math., 54 (1982), 123–201.

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[12] J. Sturm. On the congruence of modular forms. Number Theory, New York, 1984–85, Lect. Notes in Math., 1240, Springer-Verlag, 1987, Berlin, pp. 275–280.

M. Ram Murty Department of Mathematics and Statistics, Queen’s University, Jeffery Hall Kingston, Ontario K7L 3N6, Canada