Congruences between modular forms: Raising the level and dropping Euler factors

Abstract

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida’s result accounts for congruences in terms of the value of an L-function, and Ribet’s result is related to the behavior of the period that appears there. Wiles’ theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at “nonminimal level” is obtained from one at “minimal level” by dropping Euler factors from the L-function.

An example of a congruence between modular forms is provided by the newforms

of levels 11 and 77, respectively, whose first few Fourier coefficients are found in Table 1. One can show that, in fact, a_n ≡ b_n mod 3 for all n not divisible by 7. (See Theorem 5.1 below.)

We shall discuss the relationship among the following three results concerning congruences to a newform ƒ of weight 2 and level N. We assume that K is a number field containing the coefficients of ƒ and restrict our attention to congruences mod powers of a prime λ dividing ℓ.

•  A formula of Hida (1) measuring congruences to ƒ in terms of the value of an L-function.

•  A result of Ribet (2) that establishes the existence of certain systematic congruences between ƒ and forms of level Np (such as the one above).

•  A theorem of Wiles (3), completed by his work with Taylor (4), which shows that all suitable deformations of Galois representations associated to ƒ actually arise from forms congruent to ƒ.

Hida’s formula, though not part of the logical structure of ref. 3, provides some insight into the role played in Wiles’ proof by a certain generalization of Ribet’s result. This generalization can be interpreted as the invariance of a period appearing in Hida’s formula. Using this invariance, one shows that Wiles’ theorem at minimal level implies the theorem at nonminimal level.

Remark 1.1: We are concerned here mainly with Ribet’s “raising the level” result, rather than his “lowering the level” result of ref. 5. We remark that Hida also found systematic congruences between ƒ and forms of level Nℓ^r. We shall not discuss these, but focus on congruences between ƒ and forms of level Nd with d not divisible by ℓ.

Notation and Review

We fix a prime ℓ and embeddings Q̄ → Q̄_ℓ and Q̄ → C. Suppose that K is a number field contained in C and let λ denote the prime of 𝒪_K determined by our choice of embeddings. Let 𝒪 denote the localization of 𝒪_K at λ.

We suppose that ƒ is a newform of weight 2, level N_ƒ and character χ_ƒ with coefficients in K. The Eichler–Shimura construction associates to ƒ an ℓ-adic representation

such that if p does not divide N_ƒℓ, then ρ_ƒ is unramified at p and ρ_ƒ(Frob_p) has characteristic polynomial

1

We let ρ̄_ƒ denote the semisimplification of the reduction of ƒ. If ƒ and g are newforms of weight 2, then we write ƒ ∼ g if ρ̄_ƒ is equivalent to ρ̄_g. By the Cebotarev density theorem and the Brauer–Nesbitt theorem, we have ƒ ∼ g if and only if a_p(ƒ) ≡ a_p(g) for all primes p not dividing N_ƒN_gℓ, the congruence being modulo the maximal ideal of the integral closure of Z_ℓ in Q̄_ℓ.

We assume throughout that ℓ is odd, ℓ² does not divide N_ƒ, and ℓ does not divide the conductor of χ_ƒ. We assume also that the restriction of ρ̄_f to Gal (Q̄/F) is irreducible where F is the quadratic subfield of Q(ζ_ℓ). It is convenient to distinguish two sets of primes which can create technical problems.

•  We let S_ƒ denote the set of primes p such that ρ_ƒ|d_p is not minimally ramified in the sense of ref. 6.

•  We let P_ƒ denote the set of primes ρ ≠ ℓ such that ρ̄_ƒ^I_p = 0, but ad⁰(ρ̄_ƒ)^I_p ≠ 0.

If p is not in P_ƒ ∪ ℓ, then p is in S_ƒ if and only if the powers of p differ in the conductors of ρ_ƒ and ρ̄_ƒ. In the introductory example, we have S_ƒ = P_ƒ = P_g = ∅, and S_g = {7}.

Counting Congruences

We assume that N is divisible by N_ƒ but not by ℓ² and let

Let T_N denote the 𝒪-subalgebra of ∏_g∈FN C generated by the set of T_p for p not dividing Nℓ, where T_p denotes (a_p(g))_g. Then T_N is a local ring, free over 𝒪 of rank equal to the cardinality of F_N.

Consider the homomorphism π_ƒ:T_N → 𝒪 defined by projection to the ƒ coordinate. Define ideals of T_N by

Then the ideal π_ƒ(J_ƒ) has finite index in 𝒪, and is called a congruence ideal. This is a variant of the notion of a congruence module used in refs. 1 and 2.

To see how it measures congruences, consider again the above example with ƒ of level 11. We suppose that N = 77 and ℓ = 3. Then T₇₇ can be identified with

and we find that the congruence ideal is 3𝒪.

We consider also some useful variants. Suppose that Σ is a finite set of primes containing S_ƒ. We let F_Σ denote the set

We then define T_Σ as above, but using the set F_Σ instead of F_N. We denote the resulting congruence ideal C_ƒ,Σ. If ƒ is replaced by the newform associated to a twist, then T_Σ is replaced by a ring to which it is canonically isomorphic, and we obtain the same congruence ideal. So we suppose from now on that χ_ƒ is of order not divisible by ℓ.

If Σ contains P_ƒ, then F_Σ can be identified with F_NΣ for a certain integer N_Σ. Assuming this holds, we shall also associate to ƒ and Σ a cohomology congruence ideal.

Let Γ_H(N_Σ) denote the maximal subgroup of Γ₀(N_Σ) in which Γ₁(N_Σ) has ℓ-power order. Let T denote the 𝒪-subalgebra of

generated by the Hecke operators T_n for n ≥ 1. We let ƒ_Σ denote the normalized T-eigenform characterized by

•  the newform associated to ƒ_Σ is ƒ;

•  a_p(ƒ_Σ) = 0 for primes p in Σ − {ℓ};

•  a_l(ƒ_Σ) is a unit in 𝒪 if ℓ divides N_Σ;

where we have enlarged K if necessary. Consider the prime ideal θ in T defined as the kernel of the map T → 𝒪 arising from ƒ_Σ, and let m denote the maximal ideal generated by θ and λ. If ρ̄_ƒ is irreducible, the completion of T_Σ at its maximal ideal can be identified with the completion of T at m. (See section 4.2 of ref. 7.)

We now define a cohomology congruence ideal using the cohomology of the modular curve X_Σ = X_H(N_Σ) = Γ_H(N_Σ)∖ℋ*. We have a natural action of T on

We choose a basis {x, y} for the rank two submodule M = H¹(X_Σ,𝒪)[θ], the intersection of the kernels of the elements of θ. We define the cohomology congruence ideal

where 〈,〉 is the perfect pairing on H¹(X_Σ,𝒪) gotten from x∪Wy, where W is the Atkin–Lehner involution. One checks the following (see section 4.4 of ref. 7).

Lemma 3.1. The ideal C_ƒ,Σ is contained in C_ƒ,Σ^coh. Furthermore if the completion H¹(X_Σ,𝒪)_m is free over T_m≅T_Σ, then equality holds.

Remark 3.2: The freeness of H¹(X_Σ,𝒪)_m is equivalent to H¹(X_Σ,k)[m] being two-dimensional over k, which is known under our hypotheses through work of Mazur et al. (see section 2.1 of ref. 3).

Relation with L-Functions

Hida’s formula relates C_ƒ,Σ^coh to the value of an L-function. We consider the L-function associated to the Galois representation ad⁰ρ_ƒ. This L-function is defined by analytic continuation of the Euler product

2

where for primes p not dividing N_ƒ, the Euler factor L_p(ad⁰ρ_ƒ,s) is

α_p and β_p being the roots of Eq. 1. We shall not give here the recipe for the Euler factors at primes p dividing N_ƒ. We remark, however, that L(ad⁰ƒ,s) remains the same if ƒ is replaced by the newform associated to a twist, and that if N_ƒ is minimal among such newforms, then L_p(ad⁰ƒ,s) for p dividing N_ƒ is one of the following:

If Σ is a finite set of primes, then we write L^Σ(ad⁰ƒ,s) for the function obtained by omitting the Euler factors at the primes in Σ.

Suppose now that Σ contains P_ƒ∪S_ƒ as at the end of preceding section. We let ω denote the class in H¹(X_Σ,C) associated to the holomorphic differential 2πiƒ_Σ(τ)dτ on X_Σ. We let ω′ denote the class associated to the antiholomorphic differential where ω^c is defined using ƒ_Σ^c = ∑ā_n(ƒ_Σ)e^2πinτ instead of ƒ_Σ.

Viewing M as contained in H¹(X_Σ,C), we find that the span of x and y coincides with that of ω and ω′. We write A for the matrix in GL₂(C) such that

Define the period Ω to be the determinant of A. (Note that because we have chosen a basis for M, Ω is well defined only up to a unit in 𝒪.) Set δ = 3 if ℓ is in Σ, 1 if ℓ|N_ƒ but ℓ ∉ Σ, and 0 otherwise. Hida’s formula can then be stated as follows:

Theorem 4.1. C_ƒ,Σ^coh is generated by

The proof uses results of Shimura to express the Petersson inner product of ƒ with itself in terms of the value of the L-function. In particular, the ratio is an element of 𝒪.

Recall that we have assumed here that Σ contains P_ƒ∪S_ƒ, but the formula actually holds assuming only that Σ contains S_ƒ. However, we have not explained how to define Ω in that situation. We shall see that in fact

Theorem 4.2. Ω is independent of Σ.

So we could use any Σ containing S_ƒ∪P_ƒ to define Ω. From the theorem, we also see precisely how C_ƒ,Σ^coh varies with Σ: Adding primes other than ℓ to Σ simply corresponds to dropping the corresponding Euler factors from the L-function. Furthermore, we shall see that the congruences established by Ribet are related to the theorem, which is essentially a reformulation of Wiles’ generalization (3) of Ribet’s result.

Dropping Euler Factors

Ribet’s result (2) on “raising the level” is the following theorem:

Theorem 5.1. If p does not divide N_ƒ then the following are equivalent: (a) There exists g such that ƒ ∼ g, χ_{ƒ =} χ_g and N_g = dp for some divisor d of N_ƒ.

(b) The congruence a_p(ƒ)² ≡ χ_ƒ(p)(p + 1)² mod λ holds.

The introductory example is a congruence as in the theorem. We take p = 7 and λ dividing 3. Because a_p(ƒ) = −2, we see there must be a form g congruent to ƒ with N_g = 77 (because N_g = 7 is impossible).

The direction (a) ⇒ (b) of the theorem follows from consideration of the representation ρ̄_ƒ. We give the idea of the proof in the case p ≠ ℓ: If there exists a g as in the theorem, then the ratio of the eigenvalues of ρ̄_ƒ (Frob_p) must be p^±1mod λ. Then one applies the formula

The direction (b) ⇒ (a) is closely related to Theorem 4.2, which shows that

if p is not in Σ and does not divide N_ƒ. Ribet’s proof relies on a comparison of cohomology congruence ideals, but his setup is slightly different from the one here. He compares cohomology congruence ideals at level N_ƒ and N_ƒp, with the result that the factor of p − 1 does not occur.

To prove Theorem 4.2, one defines a certain T_Σ′-linear injection

It is defined so that φ(M)⊂M′ where ′ indicates we are using Σ′ instead of Σ. We may even normalize the map so that this restriction, tensored with C, sends ƒ_Σ to ƒ_Σ′, i.e., the map drops Euler factors. The key ingredient in the proof of independence is the following generalization by Wiles of a lemma of Ribet:

Lemma 5.2. φ has torsion-free cokernel.

This is proved using a result of Ihara whose role in the comparison of cohomology congruence ideals is identified in Ribet’s work.

It follows that φ induces an isomorphism M → M′, and we conclude that A = A′ using φ(x),φ(y) as a basis for M′. From Theorem 4.2 we deduce:

Corollary 5.3. Suppose that Σ′⊃Σ⊃P_ƒ∪S_ƒ. If ℓ is not in Σ′ − Σ, then let ɛ = 0. Otherwise let ɛ = 2 or 3 according to whether or not N_ƒ is divisible by ℓ. Then

3

Relation with Selmer Groups

Using Mazur’s theory of deformations of Galois representations, one associates a ring R_Σ and a universal deformation

of ρ̄_ƒ minimally ramified outside Σ (see ref. 6). Here we work over the completion 𝒪 of 𝒪̂, which we view as contained in Q̄_ℓ. Supposing that Σ contains S_ƒ, we obtain a homomorphism π_ƒ,Σ: R_Σ → Q̄_ℓ from ρ_ƒ and the universal property. The 𝒪̂-module

can be described using Galois cohomology. In fact we have a canonical isomorphism

4

where L is gotten from ad⁰ρ_ƒ. The group on the right is sometimes called a Selmer group. The subscript Σ indicates that for p∉Σ the cohomology classes are supposed to restrict to elements of H_ƒ¹(G_p,L ⊗_ZℓQ_ℓ/Z_ℓ) (as defined in ref. 8). There is also a possibly weaker condition imposed at p = ℓ if it is in Σ (3, 9). The universal property of the deformation also yields a surjective homomorphism φ_Σ from R_Σ to the completion of T_Σ. The key result of Wiles (3) and its generalization in (9) is that φ_Σ is an isomorphism (6, 7).

This result turns out to be related to the comparison of the congruence ideal C_ƒ,Σ with the Fitting ideal of Φ_ƒ,Σ, which we denote D_ƒ,Σ. (Recall that if Φ_ƒ,Σ has finite length d, then its Fitting ideal is generated by λ^d, and if the length is infinite than the Fitting ideal is trivial.) On the one hand, an easy commutative algebra argument shows that

5

On the other hand, a deeper commutative algebra argument shows that equality holds in Eq. 5 if and only if the following hold: (a) φ_Σ is an isomorphism, and (b) T_Σ is a complete intersection.

One first proves the two assertions in the case Σ = ∅, so to get started one needs the existence of ƒ such that S_ƒ = ∅. This existence is a version of Serre’s epsilon conjecture, and the most difficult step in the proof is Ribet’s theorem on lowering the level (5). Assuming that we also have P_ƒ = ∅, Taylor and Wiles (4) show that T_∅ is a complete intersection, and using this fact Wiles (3) shows that φ_∅ is an isomorphism. Their proofs use the generalization of Mazur’s result discussed in Remark 3.2, and from which we also deduce

6

if Σ = S_ƒ = P_ƒ = ∅.

Combining the inclusion Eq. 3 with its counterpart

resulting from a Galois cohomology argument, we find that Eq. 6 holds for arbitrary Σ provided S_ƒ = P_ƒ = ∅. Hence we have (a) and (b), and therefore D_ƒ,Σ = Ĉ_ƒ,Σ, assuming only that Σ ⊃ S_ƒ and P_ƒ = ∅. Applying the result of remark 3.2, we get Eq. 6 as well in that case.

Remark 6.1: Improvements to these arguments, due to Faltings, Lenstra, Fujiwara, and the author (10) establish (a), (b), and Eq. 6 simultaneously (first for Σ = ∅, then in general) without appealing to Remark 3.2.

If P_ƒ is not empty, then we can sometimes get empty P_ƒ for a twist, but in general we appeal to ref. 9 to get (a) and (b) in the case of Σ = S_ƒ = ∅, along with Eq. 3 if Σ′ = P_ƒ. We conclude that

Theorem 6.2. Keep the above hypotheses and notation.

•  For arbitrary ∑, (a) and (b) hold.

•  If ∑ contains S_f, then  is a generator for D_f,∑.

•  If ∑ contains S_f ∪ P_f, then Eq. 6 holds.

Remark 6.3: Coates and Flach have pointed out that one can deduce form the theorem a formula relating the order of H_∅¹ (G_Q,L⊗_zℓQ_ℓ/Z_ℓ) to L^Σ(ad⁰f,1). To relate the orders of H_∅¹ and H_Σ¹, one uses a variant of proposition 5.14 (ii) of ref. 8. In the case of f corresponding to an elliptic curve, see section 3 of ref. 11 for this variant and ref. 12 for a discussion of the relation with the Tamagawa number conjecture (8).

Table 1.

Fourier coefficients

n	1	2	3	4	5	6	7	8
an	1	−2	−1	2	1	2	−2	0
bn	1	1	2	−1	−2	2	−1	3