Let be a chosen algebraic closure of Q and G = Gal(/Q). For each prime number , we choose an algebraic closure of Q together with an embedding of into and we set G = Gal(/Q) 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 _p (one has Σ_rZh^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, 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, 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 , generated by Hecke operators acting on some automorphic representation space, equipped with a continuous homomorphism ρ : G → GL_d(), “compatible with the action of the Hecke operators,” such that V comes from (i.e., is isomorphic to the one we get from ρ via a map → 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.

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

Denote by the category of local noetherian complete _E-algebras with residue field k (we will simply call the objects of this category _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 of Rep_Zp^f(J) stable under subobjects, quotients, and direct sums.

For A in , 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 . The A-representations of J lying in form a full subcategory (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 . For any A in , 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(A) = F_u,J,(A) = the subset of F(A) corresponding to representations which lie in .

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

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

Fix also a flat _E-representation U of J lifting u and lying in . Its class defines an element of F(_E) F(_E), hence augmentations _U:R → _E and _U,:R → _E.

Set _n = _E/πⁿ_E and U_n = U/πⁿU. If p_U = ker _U and p_U, = ker _U,, we have canonical isomorphisms

We fix a geometric E-representation V of G (morally a “Hecke representation”). We choose a G-stable _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 _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 _p if a G_p-stable lattice lies in _p.

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

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

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

(ii) V′ is of type (S, _p).

Then, if Q_S denote the maximal Galois extension of Q contained in unramified outside of S, deformation theory applies with J = G_S = Gal(Q_S/Q) and the full subcategory of Rep_Zp^f(G_S) whose objects are T’s which, viewed as representations of G_p, are in _p. But if we want the definition of (S, _p)-close to V to be good for our purpose, it is crucial that the category _p is semistable, i.e., is such that any E-representation of G_p lying in _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 _p. Since H¹(J, gl(U_n)) is the kernel of the natural map

_p

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

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

For any _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),

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

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 _p denote the p-adic completion of the normalization of Z_p in _p, the ring

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 ^o : Fil⁰(A_cris M) → A_cris M as being ^{i −i} on Filⁱ A_cris Fil⁻ⁱ M. If we set

Proposition. Let V′ be an E-representation of G_p. Then V′ lies in _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_p, Z/pⁿZ) is an object of _p^cr(Z_p).

Remarks: (i) Define _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, _p^ff is a full subcategory stable under extensions of _p^cr (this is the essential image of MF^[−1,0](Z_p)).

(ii) Deformations in _p^cr don’t change Hodge type: if V,V′ are E-representations of G_p, lying in _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).

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

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

_E

_E

_E

Hence, if U is a G_p-stable lattice of an E-representation V of G_p lying in _p^cr, and if, for any i Z, h_r = h_r(V), with obvious notations, we get H¹_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_E(M_n)) and lg_EH¹_p^cr(Q_p,gl(U_n)) = lg_E H⁰(Q_p,gl(U_n)) + nh, where h = Σ_i<jh_ih_j [this generalizes a result of Ramakrishna (9)].

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,p^cr. In this case, H¹_p^cr(Q_p, gl(U_n)) (_n)^h+1 and H¹_p^cr(Q_p,sl(U_n)) (_n)^h. Moreover, because there is no H², the deformation problem is smooth, hence R_u,Gp,_{p^cr E}[[X₀, X₁, X₂,…,X_h]].

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

For any _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) Nⁱ = ⁱ⁻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

There is again a simple characterization of the category _p^na(E) of E-representations of G_p lying in _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 _p^na(E).

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

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

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

Again, in this case, H¹_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_E(M_n)). But,

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

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

Example 3: _p^st [the good generalization of _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) as the unique Z_p-endomorphism such that (u) = u^p;

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

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

If r ≤ p − 2, let ′₀^r be the category whose objects consist of:

(i) an S-module ,

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

(iii) a linear map ^r: Fil^r → , such that ^r(sx) = ^r(s).(x) (where : → is defined by (x) = ^r(v^rx)/^r(v^r)), with an obvious definition of the morphisms. We consider the full subcategory ₀^r of ′₀^r whose objects satisfy

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

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

Finally, define ^r as the category whose objects are objects of ₀^r equipped with a linear endomorphism

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

(ii) v.N(Fil^r) Fil^r,

(iii) if x Fil^r, ¹(v).N(^r(x)) = ^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 _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”:

Let V be an E-representation of G_p. Breuil proves that, if V lies in _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 _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_p, Z/pⁿZ) is an object of _p^st,r(Z_p).

When working with _p^st,r, deformation may change the Hodge type (the conductor also). The computation of H¹_p^st,r(Q_p, gl(U_n)) still reduces to a computation in MFB^[−r,o](_E) (or equivalently in ^r). This computation becomes difficult in general but can be done in specific examples.

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

(a) Call _p^ff,L, the full subcategory of Rep_Zp^f(G_p) whose objects are representations which, when restricted to Gal(_p/Q_p), extends to a finite and flat group scheme over _L. If e_L ≤ p − 1, an E-representation V lies in _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 _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 _p^cr. For e_L = p − 1, the same thing holds if we require that the representation of Gal(_p/Q_p) extends to a connected finite and flat group scheme over _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 _p^cris,r,L (resp. _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 _p^cris,r,L (resp. _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.

This paper was partially supported by the Institut Universitaire de France and Centre National de la Recherche Scientifique, Unité de Recherche Associée D0752.