Deforming semistable Galois representations

Abstract

Let V be a p-adic representation of Gal(Q̄/Q). One of the ideas of Wiles’s proof of FLT is that, if V is the representation associated to a suitable autromorphic form (a modular form in his case) and if V′ is another p-adic representation of Gal(Q̄/Q) “closed enough” to V, then V′ is also associated to an automorphic form. In this paper we discuss which kind of local condition at p one should require on V and V′ in order to be able to extend this part of Wiles’s methods.

Geometric Galois Representations (refs. 1 and 2; exp. III and VIII).

Let Q̄ be a chosen algebraic closure of Q and G = Gal(Q̄/Q). For each prime number ℓ, we choose an algebraic closure Q̄_ℓ of Q_ℓ together with an embedding of Q̄ into Q̄_ℓ and we set G_ℓ = Gal(Q̄_ℓ/Q_ℓ) ⊂ G. We choose a prime number p and a finite extension E of Q_p.

An E-representation of a profinite group J is a finite dimensional E vector space equipped with a linear and continuous action of J.

An E-representation V of G is said to be geometric if

(i) it is unramified outside of a finite set of primes;

(ii) it is potentially semistable at p (we will write pst for short).

[The second condition implies that V is de Rham, hence Hodge-Tate, and we can define its Hodge-Tate numbers h^r = h^r(V) = dim_E (C_p(r) ⊗_Qp V)^G_p where C_p(r) is the usual Tate twist of the p-adic completion of Q̄_p (one has Σ_r∈Zh^r = d). It implies also that one can associate to V a representation of the Weil-Deligne group of Q_p, hence a conductor N_V(p), which is a power of p].

Example: If X is a proper and smooth variety over Q and m ∈ N, j ∈ Z, then the p-adic representation H_et^m(X_Q̄, Q_p(j)) is geometric.

[Granted the smooth base change theorem, the representation is unramified outside of p and the primes of bad reduction of X. Faltings (3) has proved that the representation is crystalline at p in the good reduction case. It seems that Tsuji (4) has now proved that, in case of semistable reduction, the representation is semistable. The general case can be deduced from Tsuji’s result using de Jong’s (5) work on alterations].

Conjecture (1). If V is a geometric irreducible E-representation of G, then V comes from algebraic geometry, meaning that there exist X, m, j such that V is isomorphic, as a p-adic representation, to a subquotient of E ⊗_QpH_et^m(X_Q̄, Q_p(j)).

Even more should be true. Loosely speaking, say that a geometric irreducible E-representation V of G is a Hecke representation if there is a finite Z_p-algebra ℋ, generated by Hecke operators acting on some automorphic representation space, equipped with a continuous homomorphism ρ : G → GL_d(ℋ), “compatible with the action of the Hecke operators,” such that V comes from ℋ (i.e., is isomorphic to the one we get from ρ via a map ℋ → E). Then any geometric Hecke representation of G should come from algebraic geometry and any geometric irreducible representation should be Hecke.

At this moment, this conjecture seems out of reach. Nevertheless, for an irreducible two-dimensional representation of G, to be geometric Hecke means to be a Tate twist of a representation associated to a modular form. Such a representation is known to come from algebraic geometry. Observe that the heart of Wiles’s proof of FLT is a theorem (6, th. 0.2) asserting that, if V is a suitable geometric Hecke E-representation of dimension 2, then any geometric E-representation of G which is “close enough” to V is also Hecke.

It seems clear that Wiles’s method should apply in more general situations to prove that, starting from a suitable Hecke E-representation of G, any “close enough” geometric representation is again Hecke. The purpose of these notes is to discuss possible generalizations of the notion of “close enough” and the possibility of extending local computations in Galois cohomology which are used in Wiles’s theorem. More details should be given elsewhere.

Deformations (7–9).

Let 𝒪_E be the ring of integers of E, π a uniformizing parameter and k = 𝒪_E/π𝒪_E the residue field.

Denote by 𝒞 the category of local noetherian complete 𝒪_E-algebras with residue field k (we will simply call the objects of this category 𝒪_E-algebras).

Let J be a profinite group and Rep_Zp^f(J) the category of Z_p-modules of finite length equipped with a linear and continuous action of J. Consider a strictly full subcategory 𝒟 of Rep_Zp^f(J) stable under subobjects, quotients, and direct sums.

For A in 𝒞, an A-representation T of J is an A-module of finite type equipped with a linear and continuous action of J. We say that T lies in D if all the finite quotients of T viewed as Z_p-representations of J are objects of 𝒟. The A-representations of J lying in 𝒟 form a full subcategory 𝒟(A) of the category Rep_A^tf(J) of A-representations of J.

We say T is flat if it is flat (⇔ free) as an A-module.

Fix u a (flat !)-k-representation of J lying in 𝒟. For any A in 𝒞, let F(A) = F_u,J(A) be the set of isomorphism classes of flat A-representations T of J such that T/πT ≃ u. Set F_𝒟(A) = F_u,J,𝒟(A) = the subset of F(A) corresponding to representations which lie in 𝒟.

Proposition. If H⁰(J, gl(u)) = k and dim_kH¹(J, gl(u)) < +∞, then F and F_𝒟 are representable.

(The ring R_𝒟 = R_u,J,𝒟 which represents F_𝒟 is a quotient of the ring R = R_u,J representing F.)

Fix also a flat 𝒪_E-representation U of J lifting u and lying in 𝒟. Its class defines an element of F_𝒟(𝒪_E) ⊂ F(𝒪_E), hence augmentations ɛ_U:R → 𝒪_E and ɛ_U,𝒟:R_𝒟 → 𝒪_E.

Set 𝒪_n = 𝒪_E/πⁿ𝒪_E and U_n = U/πⁿU. If p_U = ker ɛ_U and p_U,_𝒟 = ker ɛ_U,_𝒟, we have canonical isomorphisms

Close Enough to V Representations.

We fix a geometric E-representation V of G (morally a “Hecke representation”). We choose a G-stable 𝒪_E-lattice U of V and assume u = U/πU absolutely irreducible (hence V is a fortiori absolutely irreducible).

We fix also a finite set of primes S containing p and a full subcategory 𝒟_p of Rep_Zp^f(G_p), stable under subobjects, quotients, and direct sums.

For any E-representation W of G_p, we say W lies in 𝒟_p if a G_p-stable lattice lies in 𝒟_p.

We say an E-representation of G is of type (S, 𝒟_p) if it is unramified outside of S and lies in 𝒟_p.

Now we assume V is of type (S, 𝒟_p). We say an E-representation V′ of G is (S, 𝒟_p)-close to V if:

(i) given a G-stable lattice U′ of V′, then U′/πU′ ≃ u;

(ii) V′ is of type (S, 𝒟_p).

Then, if Q_S denote the maximal Galois extension of Q contained in Q̄ unramified outside of S, deformation theory applies with J = G_S = Gal(Q_S/Q) and 𝒟 the full subcategory of Rep_Zp^f(G_S) whose objects are T’s which, viewed as representations of G_p, are in 𝒟_p. But if we want the definition of (S, 𝒟_p)-close to V to be good for our purpose, it is crucial that the category 𝒟_p is semistable, i.e., is such that any E-representation of G_p lying in 𝒟_p is pst.

We would like also to be able to say something about the conductor of an E-representation of G_p lying in 𝒟_p. Since H_𝒟¹(J, gl(U_n)) is the kernel of the natural map

it is better also if we are able to compute H_𝒟p¹(G_p, gl(U_n)).

In the rest of these notes, we will discuss some examples of such semistable categories 𝒟_p’s.

Examples of Semi-Stable 𝒟_p’s.

Example 1: The category 𝒟_p^cr (application of (10); cr, crystalline).

For any 𝒪_E-algebra A, consider the category MF(A) whose objects are A-module M of finite type equipped with

(i) a decreasing filtration (indexed by Z),

by sub-A-modules, direct summands as Z_p-modules, with Filⁱ M = M for i ≪ 0 and = 0 for i ≫ 0;

(ii) for all i ∈ Z, an A-linear map φⁱ : Filⁱ M → M, such that φⁱ |_{Filⁱ⁺¹ M}= pφⁱ⁺¹ and M = ∑Im φⁱ.

With an obvious definition of the morphisms, MF(A) is an A-linear abelian category.

For a ≤ b ∈ Z, we define MF^[a,b](A) to be the full subcategory of those M, such that Fil^a M = M and Fil^b+1 M = 0. If a < b, we define also MF^]a,b](A) as the full subcategory of MF^[a,b](A) whose objects are those M with no nonzero subobjects L with Fil^a+1 L = 0.

As full subcategories of MF(A), MF^[a,b](A) and MF^]a,b](A) are stable under taking subobjects, quotients, direct sums, and extensions.

If Z̄_p denote the p-adic completion of the normalization of Z_p in Q̄_p, the ring

is equipped with an action of G_p and a morphism of Frobenius φ : A_cris → A_cris. There is a canonical map A_cris → Z̄_p whose kernel is a divided power ideal J. Moreover, for 0 ≤ i ≤ p − 1, φ(J^[i]) ⊂ pⁱA_cris. Hence, because A_cris has no p-torsion, we can define for such an i, φⁱ : J^[i] → A_cris as being the restriction of φ to J^[i] divided out by pⁱ.

For M in MF^{[−(p−1),0]}(A), we then can define Fil^o(A_cris ⊗ M) as the sub-A-module of A_cris ⊗_Zp M, which is the sum of the images of the FilⁱA_cris ⊗ Fil⁻ⁱM, for 0 ≤ i ≤ p − 1. We can define φ^o : Fil⁰(A_cris ⊗ M) → A_cris ⊗ M as being φⁱ ⊗ φ⁻ⁱ on Filⁱ A_cris ⊗ Fil⁻ⁱ M. If we set

this is an A-module of finite type equipped with a linear and continuous action of G_p. We get in this way an A-linear functor

which is exact and faithful. Moreover, the restriction of U to MF^{]−(p−1),0]}(A) is fully faithful. We call 𝒟_p^cr(A) the essential image.

Proposition. Let V′ be an E-representation of G_p. Then V′ lies in 𝒟_p^cr if and only if the three following conditions are satisfied:

(i) V′ is crystalline (i.e., V′ is pst with conductor N_V′(p) = 1);

(ii) h^r(V′) = 0 if r > 0 or r < −p + 1;

(iii) V′ has no nonzero subobject V" with V"(−p + 1) unramified.

Moreover (11), if X is a proper and smooth variety over Q_p with good reduction and if r,n ∈ N with 0 ≤ r ≤ p − 2, H_et^r(X_Q̄p, Z/pⁿZ) is an object of 𝒟_p^cr(Z_p).

Remarks: (i) Define 𝒟_p^ff as the full subcategory of Rep_Zp^f (G_p), whose objects are representations which are isomorphic to the general fiber of a finite and flat group scheme over Z_p. If p ≠ 2, 𝒟_p^ff is a full subcategory stable under extensions of 𝒟_p^cr (this is the essential image of MF^[−1,0](Z_p)).

(ii) Deformations in 𝒟_p^cr don’t change Hodge type: if V,V′ are E-representations of G_p, lying in 𝒟_p^cr and if one can find lattices U of V and U′ of V′ such that U/πU ≃ U′/πU′, then h^r(V) = h^r(V′) for all r ∈ Z (if U/πU = U(M), h^r(V) = dim_kgr^−rM).

Computation of H_𝒟¹_p^cr.

This can be translated in terms of the category MF(𝒪_E) ⊃ MF^]−p+1,0](𝒪_E).

In MF(𝒪_E), define H_MFⁱ(Q_p, M) as being the i^th derived functor of the functor Hom_MF(𝒪E) (𝒪_E, −). These groups are the cohomology of the complex

If we set t_M = M/Fil⁰M, this implies lg_𝒪EH_𝒟¹_p^cr(Q_p, M) = lg_𝒪EH⁰ + lg_𝒪Et_M.

Hence, if U is a G_p-stable lattice of an E-representation V of G_p lying in 𝒟_p^cr, and if, for any i ∈ Z, h_r = h_r(V), with obvious notations, we get H_𝒟¹_p^cr(Q_p, gl(U_n)) = Ext_MF^1]−p+1,0]_(A)(M_n, M_n) = Ext_MF(A)¹(M_n, M_n) = H_MF¹(Q_p,End_𝒪E(M_n)) and lg_𝒪EH_𝒟¹_p^cr(Q_p,gl(U_n)) = lg_𝒪E H⁰(Q_p,gl(U_n)) + nh, where h = Σ_i<jh_ih_j [this generalizes a result of Ramakrishna (9)].

A Special Case.

Of special interest is the case where H⁰(Q_p,gl(u)) = k, which is equivalent to the representability of the functor F_u,Gp,𝒟_p^cr. In this case, H_𝒟¹_p^cr(Q_p, gl(U_n)) ≃ (𝒪_n)^h+1 and H_𝒟¹_p^cr(Q_p,sl(U_n)) ≃ (𝒪_n)^h. Moreover, because there is no H², the deformation problem is smooth, hence R_u,Gp,_𝒟p^cr ≃ 𝒪_E[[X₀, X₁, X₂,…,X_h]].

Example 2: 𝒟_p^na (the naive generalization of 𝒟_p^cr to the semistable case).

For any 𝒪_E-algebra A, we can define the category MFN(A) whose objects consist of a pair (M, N) with M object of MF(A) and N : M → M such that

(i) N(FilⁱM) ⊂ Filⁱ⁻¹M,

(ii) Nφⁱ = φⁱ⁻¹N.

With an obvious definition of the morphisms, this is an abelian A-linear category and MF(A) can be identified to the full subcategory of MFN(A) consisting of M’s with N = 0.

We have an obvious definition of the category MFN^]−p+1,0](A). There is a natural way to extend U to a functor

again exact and fully faithful. We call 𝒟_p^cr(A) the essential image.

There is again a simple characterization of the category 𝒟_p^na(E) of E-representations of G_p lying in 𝒟_p^na as a suitable full subcategory of the category of semistable representations with crystalline semisimplification. Moreover:

If p ≠ 2, the category of semistable V values with h^r(V) = 0 if r ∉ {0, −1} is a full subcategory stable under extensions of 𝒟_p^na(E).

For 0 ≤ r < p − 1, let 𝒟_p^ord,r the full subcategory of Rep_Zp^f(G_p) of T’s such that there is a filtration (necessarily unique)

such that gr_iT(−i) is unramified for all i; then 𝒟_p^ord,r is a full subcategory of 𝒟_p^na stable under extensions.

Again, in 𝒟_p^na, deformations don’t change Hodge type. The conductor may change.

Computation of H_𝒟¹_p^na(Q_p, gl(U_n)).

As before, this can be translated in terms of the category MFN(𝒪_E) ⊃ MFN^]−p+1,0](𝒪_E): if we define H_MFNⁱ(Q_p, M) as the i^th-derived functor, in the category MFN(𝒪_E), of the functor Hom_MFN(𝒪E)(𝒪_E, −), these groups are the cohomology of the complex

(with x ↦ (Nx, (1 − φ⁰)x) and (y, z) ↦ (1 − φ⁻¹)y − Nz).

Again, in this case, H_𝒟¹_p^na(Q_p, gl(U_n)) = Ext_MFN^1]−p+1,0]_(A)(M_n, M_n) = Ext_MFN(A)¹(M_n, M_n) = H_MFN¹(Q_p, End_𝒪E(M_n)). But,

(i) the formula for the length is more complicated, and

(ii) the (local) deformation problem is not always smooth.

Example 3: 𝒟_p^st [the good generalization of 𝒟_p^cr to the semistable case, theory due to Breuil (12)].

Let S = Z_p<u> be the divided power polynomial algebra in one variable u with coefficients in Z_p. If v = u − p, we have also S = Z_p<v>. Define:

(a) FilⁱS as the ideal of S generated by the v^m/m!, for m ≥ i;

(b) φ as the unique Z_p-endomorphism such that φ(u) = u^p;

(c) N as the unique Z_p-derivation from S to S such that N(u) = −u.

For r ≤ p − 1, φ^r: Fil^rS → S is defined by φ^r(x) = p^−rφ(x).

If r ≤ p − 2, let ′ℳ₀^r be the category whose objects consist of:

(i) an S-module ℳ,

(ii) a sub-S-module Fil^rℳ of ℳ containing Fil^rS.ℳ,

(iii) a linear map φ^r: Fil^rℳ → ℳ, such that φ^r(sx) = φ^r(s).φ(x) (where φ: ℳ → ℳ is defined by φ(x) = φ^r(v^rx)/φ^r(v^r)), with an obvious definition of the morphisms. We consider the full subcategory ℳ₀^r of ′ℳ₀^r whose objects satisfy

(i) as an S-module ℳ ≃ ⊕_1≤i≤dS/p^n_dS for suitable integers d and (n_i)_1≤i≤d;

(ii) as an S-module ℳ is generated by the image of φ^r.

Finally, define ℳ^r as the category whose objects are objects ℳ of ℳ₀^r equipped with a linear endomorphism

satisfying

(i) N(sx) = N(s).x + s.N(x) for s ∈ S, x ∈ ℳ,

(ii) v.N(Fil^rℳ) ⊂ Fil^rℳ,

(iii) if x ∈ Fil^rℳ, φ¹(v).N(φ^r(x)) = φ^r(v.N(x)).

This turns out to be an abelian Z_p-linear category and we call MFB^[−r,o](Z_p) the opposite category.

For A an 𝒪_E-algebra, one can define in a natural way the category MFB^[−r,o](A) (for instance, if A is artinian, an object of this category is just an object of MFB^[−r,o](Z_p) equipped with an homomorphism of A into the ring of the endomorphisms of this object).

Breuil defines natural “inclusions”:

Moreover, the simple objects of MF^[−r,o](k), MFN^[−r,o](k), and MFB^[−r,o](k) are the same. Breuil extends U to MFB^[−r,o](A) and proves that this functor is again exact and fully faithful. We call 𝒟_p^st,r(A) the essential image.

Let V be an E-representation of G_p. Breuil proves that, if V lies in 𝒟_p^st,r then V is semistable and h^m(V) = 0 if m>0 or m < −r. Conversely, it seems likely that if V satisfies these two conditions, V lies in 𝒟_p^st,r. This is true if r = 1, and it has been proven by Breuil if E = Q_p and V is of dimension 2. More importantly, Breuil proved also

Proposition (13). Let X be a proper and smooth variety over Q_p. Assume X as semistable reduction and let r, n ∈ N with 0 ≤ r ≤ p−2; then H_et^r(X_Q̄p, Z/pⁿZ) is an object of 𝒟_p^st,r(Z_p).

When working with 𝒟_p^st,r, deformation may change the Hodge type (the conductor also). The computation of H_𝒟¹_p^st,r(Q_p, gl(U_n)) still reduces to a computation in MFB^[−r,o](𝒪_E) (or equivalently in ℳ^r). This computation becomes difficult in general but can be done in specific examples.

Final Remarks.

Let L be a finite Galois extension of Q_p contained in Q̄_p, 𝒪_L the ring of integers and e_L = e_L/𝒟_p.

(a) Call 𝒟_p^ff,L, the full subcategory of Rep_Zp^f(G_p) whose objects are representations which, when restricted to Gal(Q̄_p/Q_p), extends to a finite and flat group scheme over 𝒪_L. If e_L ≤ p − 1, an E-representation V lies in 𝒟_p if and only if it becomes crystalline over L and h^m(V) = 0 for m ∉ {0, −1}). If e_L < p − 1, Conrad (14) defines an equivalence between 𝒟_p^ff,L and a nice category of filtered modules equipped with a Frobenius and an action of Gal(L/Q_p). Using it, one can get the same kind of results as we described for 𝒟_p^cr. For e_L = p − 1, the same thing holds if we require that the representation of Gal(Q̄_p/Q_p) extends to a connected finite and flat group scheme over 𝒪_L.

(b) More generally, Breuil’s construction should extend to E-representations becoming semistable over L with h^m(V) = 0 if m > 0 or < −(p − 1)/e_L (≤ −(p − 1)/e_L with a “grain de sel”).

(c) Let RepQ_p(G_p)_cris,L^r (resp. RepQ_p(G_p)_st,L^r) be the category of Q_p-representations V of G_p becoming crystalline over L (resp. semistable) with h^m(V) = 0 if m > 0 or m < −r. Let 𝒟_p^cris,r,L (resp. 𝒟_p^st,r,L) be the full subcategory of Rep_Zp^f(G_p) consisting of T’s for which one can find an object V of RepQ_p(G_p)_cris,L^r (resp. RepQ_p(G_p)_st,L^r) G_p-stable lattices U′ ⊂ U of V such that T ≃ U/U′. I feel unhappy not being able to prove the following:

Conjecture. C_p^cris,r,L (resp. C_p^st,r,L): Let V be a Q_p-representation of V lying in 𝒟_p^cris,r,L (resp. 𝒟_p^st,r,L). Then V an object of Rep_Qp(G_p)_cris,L^r (resp. Rep_Qp(G_p)_st,L^r).

The only cases I know C_p^cris,r,L are r = 0, r = 1, and e_L ≤ p − 1, r ≤ p − 1, and e_L = 1. The only cases I know C_p^st,r,L are r = 0, r = 1, and e_L ≤ p − 1. Of course, each time we know the answer is yes, this implies that the category is semistable.