Adjoint modular Galois representations and their Selmer groups

1. Selmer Groups

Let G be the Galois group of the maximal extension Q ^(p)/Q unramified outside {p, ∞}. Let 𝒪 be a valuation ring finite flat over Z _p with residue field F. We start with a two-dimensional continuous representation φ : G → GL ₂(A) for a complete (noetherian) local 𝒪-algebra A with residue field F = A/m _A. The power series ring 𝒪[[T ₁, … , T _r]] is an example of such A. We let G act on V = A ² via φ and on End(V) by conjugation: φ ⊗ φ^∨ (σ)x = φ(σ)xφ(σ)⁻¹. We look at its three-dimensional factor ad(φ) : G → GL ₃(A) acting on trace zero subspace V(ad(φ)) in End(V). Thus φ ⊗ φ^∨ = ad(φ) ⊕ 1. Let φ̄ = φ mod m _A. We assume the following three conditions:

 (AI)  The restriction of φ̄ to Gal (Q ^(p)/Q( is absolutely irreducible;

(Ord)  For each decomposition group D over p, φ|_D ≅ (₀ ^δ  _ɛ ^∗) with unramified δ;

(Reg)  δ mod m_A ≠ ɛ mod m_A.

Condition AI is equivalent to the absolute irreducibility of ad(φ̄) over G. We write V(δ) ⊂ V for the δ-eigen subspace, and for each A-submodule X of V(ad(φ)), let X* = X ⊗_A A* for the Pontryagin dual A* = Hom_𝒪(A, Q _p/Z _p) of A. We put V ₊ = {ξ ∈ V(ad(φ)) ⊂ End(V) | ξ(V(δ)) = 0}. Then we define the Selmer group for ad(φ), as a special case of Greenberg’s definition (ref. 1; see also ref. 2): for the inertia subgroup I of D. This is a generalization of the class group; for example, taking a quadratic character χ of G, is the χ-part of the p-class group of the quadratic extension F fixed by ker(χ). Thus if A = 𝒪 and if L(1, ad(φ)) ≠ 0, a naive guess is that Sel(ad(φ)) is finite and that its order is the p-part of L(1, ad(φ)) up to a transcendental factor. The finiteness is first shown by Flach (3) and then by Wiles (4). We discuss later some good cases where this guess works well. We generalize the above definition to a tensor product ad(φ) ⊗ ɛ with a character ɛ : G → B ^× for a complete noetherian 𝒪-algebra B, replacing A by A⊗̂_𝒪 B and V ₊ by V ₊(ad(φ) ⊗ ɛ) = V ₊⊗̂ B: which is a discrete module over A⊗̂_𝒪 B.

2. Elliptic Curves over Q

For simplicity, we suppose that φ₀ is the Galois representation on H ¹(E _/Q̄, Z_p) for a modular elliptic curve E _/Q inside the Jacobian J = J ₀(p) of the modular curve X ₀(p). Thus E has multiplicative reduction at p and has good reduction outside p. Taking the dual of the inclusion E ⊂ J, we have a quotient map π : J → E. Then J = E + A for A = ker(π), and E ∩ A is a finite group of square order. For a Néron differential ω on the Néron model E _/Z, by a result of Mazur (5) corollary 4.1, we may assume that π*ω = 2^e(2πif ₀(z)dz) for a primitive form f ₀ ∈ S ₂(Γ₀(p)) and e ∈ Z. Choosing a base c _± of ±-eigenspace of H ₁(E(C), Z) under complex conjugation, we define Ω_± by ∫_c± ω after normalizing c _± as described below. The following formula was proven 15 years ago in ref. 6 (see also ref. 7): where C = 2^a+2e p(p − 1) for 2^a = [H ₁(E(C), Z) : Z c ₊ ⊕ Zc ₋]. We define the canonical period U(f ₀) of f ₀ by C ⁻¹(2πi)Ω₊Ω₋. In ref. 6, to get formula IN1, we used the period determinant for a Z-base {c ₁, c ₂} of H ₁(E(C), Z) in place of Ω₊Ω₋ (see ref. 6, formula 6.20b). Writing ω_± = (ω ± ω̄)/2, we see ∫_c± ω = ± ∫_c± ω_±, and thus Ω₊ ∈ R and Ω₋ ∈ R. Replacing c _± by their negative if necessary, we may assume that Ω₊ > 0 and √Ω₋ > 0. Under this normalization, formula IN1 is correct. Then by definition, 2^a u = Ω₊Ω₋, and we can deduce formula IN1 from ref. 6, theorem 6.1, by just remarking that L*_f0/L _f0 ≅ E ∩ A under the notation of the theorem quoted.

Actually, a formula similar to formula IN1 is proven in ref. 6 for the Galois representation attached to any holomorphic primitive form of weight ≥2. The formula is generalized later to cohomological cusp forms on GL(2) over imaginary quadratic fields in ref. 8.

Let H be the subalgebra of End(J) generated by Hecke operators T(n). Then π induces the projection λ : H → Z ⊂ End(E) and another projection λ′ : H → End(A). Then we define two finite modules: It is proven in ref. 7 (equation 5.8b) that as H modules. Note that Spec(C ₀) is the scheme theoretic intersection of Spec(Im(λ)) and Spec(Im(λ′)) in Spec(H). Thus we get Recently, Taylor and Wiles (4, 9) have shown that |C _0,p| = |C _1,p|, and Wiles (4) has shown This formula is a key to Wiles’ proof of Fermat’s last theorem. The fact that Sel(ad(φ₀)) has a natural map into C_1,p was first discovered by Mazur through his deformation theory of Galois representations (10). The above formula is conjectured in ref. 11 after proving the surjectivity of the map besides other relevant results.

Anyway, under the various assumptions on p that we made, we finally get a formula for the order of Sel(ad(φ₀)):

3. One-Variable Case

The cusp form f ₀ ∈ S ₂(Γ₀(p)) can be lifted to a p-adic family of p-ordinary common eigenforms f _k = Σ_n=1 ^∞ a(n; f _k)q ⁿ ∈ S _k+2(Γ₀(p), ω^−k) (k ≥ 0) for the Teichmüller character ω (cf. ref. 12, chapter 7, theorem 7.3.7). For this, we need to fix an embedding i _p : Q̄ ↪ Q̄ _p. Then “p-ordinarity” of f _k implies that the pth coefficient of f _k in its q-expansion satisfies |a(p; f _k)|_p = 1. Note that, by the multiplicative reduction hypothesis, a(p; f ₀) = ±1. This family yields a Galois representation φ : G → GL ₂ (Λ) for a finite flat 𝒪[[T]]-algebra Λ (ref. 12, section 7.5). For simplicity, we assume Λ = 𝒪[[T]]. Then writing as φ_k the specialization of φ via 1 + T ↦ u ^k for u = 1 + p, φ_k is the Galois representation of the cusp form f _k. Then the Pontryagin dual Sel*(ad(φ)) of Sel(ad(φ)) is shown by Wiles and Taylor to be a torsion 𝒪[[T]]-module of finite type, and its characteristic power series is given by the characteristic power series of the Λ-adic congruence module C _0,Λ.

Before giving the definition of C _0,Λ, we note that we have taken cohomological formulation of Galois representations. In this paper, we characterize Galois representations by the characteristic polynomial of geometric Frobenii Frob_q at primes q ≠ p. For example, φ_k is characterized by This normalization is dual to the one taken in ref. 4, but it is all right for our purpose because ad(φ_k) = ad(φ_k ^∨).

To define C_0,Λ, we need to introduce the space S _Λ of p-ordinary Λ-adic cusp forms. For that, we consider the subspace S _k+2(Γ₀(p), ω^−k; Q̄) of S _k+2(Γ₀(p), ω^−k) made of cusp forms f with a(n; f) ∈ Q̄ for all n. We consider the Q̄ _p-span S _k+2(Γ₀(p), ω^−k; Q̄ _p) of S _k+2(Γ₀(p), ω^−k; Q̄) in Q̄ _p[[q]] via q-expansion. We write S _k+2 ^ord(Γ₀(p), ω^−k; Q̄ _p) for the subspace of S _k+2(Γ₀(p), ω^−k; Q̄ _p) spanned by all p-ordinary eigenforms. An element ℱ ∈ S _Λ is a formal q-expansion Σ_n=1 ^∞ a _n(T)q ⁿ ∈ Λ[[q]] such that the specialization ℱ_k via 1 + T ↦ u ^k is the q-expansion of an element in S _k+2 ^ord(Γ₀(p), ω^−k; Q̄ _p) for all k ≥ 0. Then S _Λ is free of finite rank over Λ on which Hecke operators T(n) naturally act (ref. 12, section 7.3). Hereafter we write ℱ for the unique Λ-adic form such that ℱ_k = f _k for all k ≥ 0. Let H be the Λ-subalgebra of End_Λ(S _Λ) generated by T(n) for all n, and define a Λ-algebra homomorphism λ : H → Λ by ℱ|h = λ(h)ℱ. We also have another λ′ of H into End_Λ(ker(λ)) given by multiplication by h ∈ H on ker(λ). Then we define Then it is easy to see that C_0,Λ ≅ Λ/(η(T)) for an element η(T) ∈ Λ. We can deduce from the result of Wiles and Taylor in ref. 4 (theorem 3.3) and ref. 9 that Here the characteristic ideal char_A(M) for a torsion A-module of finite type M over a normal noetherian ring A is given by the product of prime divisors P in A with exponent given by length_AP M _P of the localization M _P at P. Note that, as shown in ref. 7 (theorem 0.1), for a canonical period U(f _k) associated to f _k, up to p-adic units. This formula is not completely satisfactory, because the p-adic L-function η(T) is determined only up to units in Λ. For Λ-adic forms of CM type, we can choose a suitable Katz p-adic L-function in place of η (11, 13–15). In general, we can only make a conjecture on the existence of a canonical p-adic L-function L _p(ad(φ)) with precise interpolation property (16), which generates char_Λ(Sel*(ad(φ))) = (η(T)) after extending scalar to the p-adic integer ring 𝒪_Ω of the p-adic completion Ω of Q̄ _p.

4. Two-Variable Case

Now we look at the universal character ν : G → 𝒪[[S]]^× deforming the identity character of G. As already said, our formulation is cohomological, and hence ν(Frob_q) = qω(q)⁻¹ for geometric Frobenius Frob_q. Writing Q _∞ for the cyclotomic Z _p-extension of Q and Γ = Gal(Q _∞/Q), the tautological character: Γ ↪ 𝒪[[Γ]] induces the above ν for S = γ − 1 for a generator γ of Γ. Then we consider Sel*(ad(φ) ⊗ ν⁻¹), which is a module over 𝒪[[T, S]] of finite type (1). Classically, the Selmer group involving the cyclotomic variable S is defined in terms of cohomology groups over the cyclotomic Z _p tower Q _∞. As shown by Greenberg (ref. 1, proposition 3.2; see also ref. 2, section 3.1), our Selmer group Sel(ad(φ) ⊗ ν⁻¹) over Q is isomorphic to the classical one over Q _∞. Recently, we have proven a control theorem for Sel(ad(φ) ⊗ ν⁻¹) giving the following theorem.

Theorem 1. The module Sel*(ad(φ) ⊗ ν⁻¹) is a torsion 𝒪[[T, S]]-module of finite type. Moreover, the characteristic power series of Sel*(ad(φ) ⊗ ν⁻¹) is of the form SΨ(T, S) in 𝒪[[T, S]] and Ψ(T, 0)|η(T) da/dT (T) in 𝒪[[T]], where a(T) is the eigen value of T(p) for ℱ lifting f ₀ (2).

In early 1980s, we constructed (17) a two-variable p-adic L-function L(T, S) in η(T)⁻¹ S𝒪[[T, S]] such that for even m with −k ≤ m ≤ 0, for a factor E like an Euler p-factor and a simple constant ∗. This L-function ηL again has ambiguity by units in Λ, although L(T, S) is uniquely determined. In ref. 16, the existence of a canonical p-adic L-functions L _p(ad(φ) ⊗ ν⁻¹) in 𝒪[[T, S]] [for ad(φ) ⊗ ν⁻¹] with precise interpolation property is conjectured. In particular, we should have an equality: Anyway, the denominator and the numerator are not yet known to exist in general in spite of the known existence of the ratio L(T, S). Because of this, we need to use η(T) as a replacement of L _p(ad(φ)).

Theorem 2. (R. Greenberg and J. Tilouine). Write ηL(T, S) = SΦ(T, S). We have We know that da/dT (0) ≠ 0 by the theorem of St. Etienne (18) due to four people at St. Etienne in France. Thus if one can prove the divisibility Φ|Ψ in 𝒪[[T, S]], the following conjecture follows.

Main Conjecture. We have Φ = Ψ up to a unit in 𝒪[[T, S]].

Actually this conjecture is close to being proven, assuming the following ordinarity conjecture on the local structure of Weissauer’s Galois representations, as discussed in the lectures of E. Urban at the Mehta Research Institute (Allahabad, India). Let us explain Urban’s strategy. First of all, there is a theory of (nearly) p-ordinary 𝒪[[T, S]]-adic forms on GSp(4), developed mainly by Tilouine and Urban (19, 20). A cohomological Hecke eigenform f on GSp(4)_/Q is called nearly p-ordinary if its eigenvalues for two standard Hecke operators at p are p-adic units under the fixed embedding Q̄ into Q̄ _p. Here the word cohomological means that the system of Hecke eigenvalues for f appears in the middle cohomology H ³ with coefficients in a polynomial representation L of a Siegel modular variety for GL(4)_/Q. In other words, f belongs to a discrete series representation whose Harish–Chandra parameter is the sum of the highest weight of L and the half sum of positive roots. For each cohomological eigenform f, Weissauer has attached a p-adic modular Galois representation ρ_f into GL(4) with characteristic polynomials of Frobenii outside p given by the Hecke polynomial (see ref. 21). Here is the ordinarity conjecture for the Galois representation.

Ordinarity Conjecture. Assume that f is nearly p-ordinary. Then the image of the decomposition group at p of ρ_f is in a Borel subgroup of GSp(4).

Weissauer’s construction gave a compatible system of l-adic representations attached to f, and ρ_f is one of its members. When ρ_f is crystalline, we have two characteristic polynomials at p. One is that of the crystalline Frobenius L _cris(X), and the other, L _et(X), is that of the Frobenius at p of a non-p-adic member of the compatible system. The p-ordinarity conjecture follows in this case if one can prove L _cris(X) = L _et(X), which is a standard conjecture and is known to be true at least for constant sheaves (that is, so to speak, the weight 0 case).

It is enough to prove the ordinarity conjecture for crystalline ρ_f for the following reason. We can glue Weissauer’s Galois representations by means of Taylor’s pseudorepresentations and attach to each 𝒪[[T, S]]-adic eigen cusp form 𝒢 a Galois representation ρ_𝒢 : G → GL ₄(F _𝕀) for the field of fractions F _𝕀 of a finite extension 𝕀 of 𝒪[[T, S]]. Thus at densely populated points on Spec(𝕀), ρ_𝒢 specializes into Weissauer’s Galois representations. Furthermore, ρ_𝒢 has densely populated specializations on Spec(𝕀) which are crystalline at p. Thus if one can prove the p-ordinarity conjecture for crystalline specializations, the image under ρ_𝒢 of each decomposition group at p is in a Borel subgroup in GSp(4), and hence the ordinarity conjecture for all specializations follows.

We now come back to the strategy for a proof of the Main Conjecture. We look at the Klingen-style 𝒪[[T, S]]-adic Eisenstein series ℰ induced from the Λ-adic form ℱ. The Galois representation ρ_ℰ attached to ℰ has values in the standard maximal parabolic subgroup, that is, it is of the following form: The constant term of ℰ at the nonstandard parabolic subgroup ℙ is almost equal to ℱ times η(T)L(T, S). Here we mean by nonstandard the parabolic subgroup given by Thus the Eisenstein ideal Eis giving congruence between ℰ and another 𝒪[[T, S]]-adic cusp form 𝒢 should be generated by η(T)L(T, S). In particular, under the p-ordinarity conjecture, Urban has shown for such Eisenstein primes P dividing Φ(T, S), if 𝒢 ≡ ℰ mod P for a cusp form 𝒢, ρ_𝒢 has values in GSp(4) and is irreducible. It was a nontrivial task to prove this because the representation is residually reducible. We also note that, to prove this, we again need the result of Wiles (4) proving the conjecture in ref. 11. The fact that ρ_𝒢 has values in GSp(4) is essential in the proof because it guarantees that the adjoint action of ρ_ℰ on the unipotent radical of the standard maximal parabolic subgroup is actually isomorphic to ad(φ) ⊗ ν⁻¹. The extension of νdet(φ) ⊗^tφ⁻¹ mod P by φ mod P induced from ρ_𝒢 can be made nonsplit because of the irreducibility of ρ_𝒢. This nontrivial extension gives rise to a nontrivial cocycle in Sel(ad(φ) ⊗ ν⁻¹) under the Ordinarity Conjecture. This is a GSp(4) version of an argument of Wiles in (22) applied to GL(2). Since it is true for each height one prime P dividing Eis, we conclude that the Eisenstein ideal Eis of ℰ divides Ψ, assuming the Ordinarity Conjecture. To establish the divisibility Φ|Eis, in other words, to establish the congruence 𝒢 ≡ ℰ mod P ^e for P ^e|Φ, we need to have precise information on ℰ (not just its existence), for example, its Fourier coefficients, its Whittaker model, and so on.