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

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 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 _ƒ N _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 T ₇₇ 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 = H ¹(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 H ¹(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 H ¹(X _Σ,𝒪)_m is free over T _m≅T _Σ , then equality holds.

Remark 3.2: The freeness of H ¹(X _Σ,𝒪)_m is equivalent to H ¹(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 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(ad ⁰ƒ,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 H ¹(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 ^{2π in τ} instead of ƒ_Σ.

Viewing M as contained in H ¹(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 GL ₂(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 _ƒ p, 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

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 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 _ƒ ¹(G _p,L ⊗_Zℓ Q _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 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 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⊗ _zℓ Q _ℓ/Z _ℓ) to L ^Σ(ad⁰ f,1). To relate the orders of H _∅ ¹ and H _Σ ¹, 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).