Deforming Semistable Galois Representations

Deforming Semistable Galois RepresentationsAuthor(s): Jean-Marc FontaineSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 94, No. 21 (Oct. 14, 1997), pp. 11138-11141Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/43246 .

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Proc. Natl. Acad. Sci. USA Vol. 94, pp. 11138-11141, October 1997 Colloquium Paper

This paper was presented at a colloquium entitled "Elliptic and Karl Rubin, held March 15-17, 1996, at the National

Deforming semistable Galois rep JEAN-MARC FONTAINE

Universit6 de Paris-Sud, Math6matique, Batiment 425, F-91405 Orsay Cedex, F

ABSTRACT Let V be a p-adic representation of Gal(Q/ Q). One of the ideas of Wiles's proof of FLT is that, if Vis the representation associated to a suitable autromorphic form (a modular form in his case) and if V' is another p-adic representation of Gal(Q/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 Vand 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 Q be a chosen algebraic closure of Q and G =

Gal(Q/Q). For each prime number (, we choose an algebraic closure Qe of Qe together with an embedding of Q into Qe and we set Ge = Gal(Qe/Qe) C G. We choose a prime number p and a finite extension E of Qp.

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 hr =

hr(V) = dimE (Cp(r) (Qp V)GP where Cp(r) is the usual Tate twist of the p-adic completion of Qp (one has Erczhr = d). It implies also that one can associate to V a representation of the Weil-Deligne group of Qp, hence a conductor Nv(p), which is a power of p].

Example: If X is a proper and smooth variety over Q and m E N, j F Z, then the p-adic representation Hemt(X, Qp(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 atp 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 ?)Q Hemt(X, Qp(/)).

Even more should be true. Loosely speaiing, say that a geometric irreducible E-representation V of G is a Hecke representation if there is a finite Zp-algebra X, generated by Hecke operators acting on some automorphic representation space, equipped with a continuous homomorphism p: G -> GLd(f), "compatible with the action of the Hecke operators," such that Vcomes from f (i.e., is isomorphic to the one we get from p via a map C -,> E). Then any geometric Hecke

? 1997 by The National Academy of Sciences 0027-8424/97/9411138-4$2.00/0 PNAS is available online at http://www.pnas.org.

I

Curves and Modular Forms, " organized by Barry Mazur 4cademy of Sciences, Washington, DC.

resentations

rance

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. Nev- ertheless, 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 Vis 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 CE be the ring of integers of E, Xr a uniformizing parameter and k = CE/wTTE the residue field.

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

Let J be a profinite group and Repfz (J) the category of Zp-modules of finite length equipped with a linear and continuous action of J. Consider a strictly full subcategory 9 of Repfz (J) stable under subobjects, quotients, and direct sums.

ForA in %, anA-representation T of J is anA-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 Zp-representations of J are objects of 0. TheA-representations of J lying in 20 form a full subcategory a0(A) of the category RepA(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 Q0. For anyA in

%, let F(A) = Fu,J(A) be the set of isomorphism classes of flat A-representations T of J such that T/rT - u. Set F2(A) = Fu,j,a(A) = the subset of F(A) corresponding to representations which lie in 50.

PROPOSITION. IfHO(J, gt(u)) = kand dimkH'(J, gt(u)) < +%o, then F and Fg are representable.

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

Fix also a flat CE-representation U of J lifting u and lying in Q0. Its class defines an element of Fg(CE) C F(CE), hence augmentations eU:R --> CE and eu,':Rg --> CE.

Set On = CE/n0CE and Un = U/T U. If pu = ker u and pu, = ker eU,G, we have canonical isomorphisms

((1pu+ rnR)/(]Pu+7rnR))*=Ext~n[^](Un, Un)Hl(J, gU[(Un))

U U U

(Pv,~ +*rnR~)/(P2,~ + wnR~)*=Ext~,,@(Vn, U,U)=: Hi(J, qU(V))

Close Enough to V Representations. We fix a geometric E-representation Vof G (morally a "Hecke representation").

1138



Colloquium Paper: Fontaine

We choose a G-stable CE-lattice U of V and assume u = U/7rU absolutely irreducible (hence Vis a fortiori absolutely irreducible).

We fix also a finite set of primes S containing p and a full

subcategory 2p of Repfzp(Gp), stable under subobjects, quotients, and direct sums.

For any E-representation W of Gp, we say W lies in 2p if a

Gp-stable lattice lies in )p. We say an E-representation of G is of type (S, 2bp) if it is

unramified outside of S and lies in Q)p. Now we assume V is of type (S, 2p). We say an E-

representation V' of G is (S, 2p)-close to V if:

(i) given a G-stable lattice U' of V', then U'/rU' = u; (ii) V' is of type (S, 92p). Then, if Qs denote the maximal Galois extension of Q con-

tained in Q unramified outside of S, deformation theory applies withJ = Gs = Gal(Qs/Q) and 2 the full subcategory of Repfz (Gs) whose objects are Ts which, viewed as representations of Gp, are in %p. But if we want the definition of (S, Qp)-close to Vto be good for our purpose, it is crucial that the category 2p is semistable, i.e., is such that any E-representation of Gp lying in 2bp is pst.

We would like also to be able to say something about the conductor of an E-representation of Gp lying in 2bp. Since

H~a(J, gf(Un)) is the kernel of the natural map

H\(Gs, lgt(Un))->Hlp(Gp, gl(Un)),

it is better also if we are able to compute Hsp(Gp, gl(Un)). In the rest of these notes, we will discuss some examples of

such semistable categories 50p'S. Examples of Semi-Stable Qp's. Example 1: The category 2pr (application of (10); cr, crys-

talline). For any CE-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),

...Fili M D Fili+l M D ....

by sub-A-modules, direct summands as Zp-modules, with Fil' M = M for i << 0 and = 0 for i >> 0;

(ii) for all i E Z, an A-linear map )i : Fili M --> M, such that i Fili+1 M= p4)i+l and M = EIm i.

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

For a < b E Z, we define MF[a,b](A) to be the full

subcategory of those M, such that Fila M = M and Filb+l M = 0. If a < b, we define also MF]ab](A) as the full subcategory of MF[a"b](A) whose objects are those M with no nonzero

subobjects L with Fila?+ 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 Zp denote the p-adic completion of the normalization of Zp in Qp, the ring

Acris = lim Ho((Spec(Zp/p)/W,)crys, struct.sheaf)

is equipped with an action of Gp and a morphism of Frobenius 4) : Acris --> Acris. There is a canonical map Acris --> Zp whose

kernel is a divided power ideal J. Moreover, for 0 < i <

p - 1, ()(J[i) C piAcris. Hence, because Acris has no p-torsion, we can define for such an i,

i

: Jil -> Acris as being the restriction of 4) to J[il divided out by pi.

ForM in MF[-(P-1)0](A), we then can define Fil?(Acris ? M) as the sub-A-module of Acris ?zp M, which is the sum of the images of the FilAMcris 0 Fil-i'M, for 0 ' i ' p - 1. We can define

d)?: Fil?(Acris M) -- >Acris ? M as being 4i 0 4-i on Fili Acris ? Fil-i M. If we set

Proc. Natl. Acad. Sci. USA 94 (1997) 11139

U(M) = (Filo(Acris 0() M))4o=l,

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

U: MF[-(P-1',0(A) --> Repft (Gp)

which is exact and faithful. Moreover, the restriction of U to MFI-(p-1)0](A) is fully faithful. We call 2'pr(A) the essential image.

PROPOSITION. Let V' be an E-representation of Gp. Then V' lies in 2p, if and only if the three following conditions are satisfied:

(i) V' is crystalline (i.e., V' is pst with conductor Nv'(p) = 1); (ii) hr(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 Qp with

good reduction and if r,n E N with 0 - r -p - 2, Hret(XQp, Z/pnZ) is an object of %pr(Zp).

Remarks: (i) Define 2pf as the full subcategory of Repfzp (Gp), whose objects are representations which are isomorphic to the general fiber of a finite and flat group scheme over Zp. If p = 2, fp? is a full subcategory stable under extensions of cpc (this is the essential image of MF[- ?l(Zp)).

(ii) Deformations in cpr don't change Hodge type: if V,V' are E-representations of Gp, lying in 2pr and if one can find lattices U of V and U' of V' such that U/rrU -- U'/IrU', then hr(V) =

hr(V') for all r C Z (if U/IrU = U(M), hr(V) = dimkgr-rM). Computation of Hecr. This can be translated in terms of the

category MF(CE) D 1F]-P+10](CE). In MF(CE), define /'MF(Qp, M) as being the ith derived

functor of the functor HomMeF(E) (CE, -). These groups are the

cohomology of the complex

Fil? M -> >M->O- ->--> ...

If we set tM = M/Fil?M, this implies lgoEHfcr(Qp, M) = lgopHO + lgEtM'

Hence, if U is a Gp-stable lattice of an E-representation Vof

Gp lying in Qpr, and if, for any i E Z, hr = hr(V), with obvious notations, we get Hecr(Qp, gt(Un)) = Ext 1-P+1,0l (A)(Mn, Mn) = ExtMF(A)(Mn, Mn) = H F(Qp,Endo(Mn)) and lgeHcr(Qp,qg(Un)) = lgE HO(Qp,gf(Un)) + nh, where h =

Li<jhihj [this generalizes a result of Ramakrishna (9)]. A Special Case. Of special interest is the case where

H?(Qp,gl(u)) = k, which is equivalent to the representability of the functor Fu C Gcr. In this case, Hlcr(Qp, gt(Un)) - (On)h and Hg1cr(Qp,?>(Un5) = (On)h. Moreover, because there is no H2, the deformation problem is smooth, hence RU,Gpgp -- CE[[X0, Xl, X2, .,Xh]].

Example 2: 2na (the naive generalization of 7pr to the semistable case).

For any CE-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) C Fili-lM, (ii) N4i = Oi- N. 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 Al's with N = 0.

We have an obvious definition of the category MFN]-p+ 0](A). There is a natural way to extend Uto a functor

U : MFNl-p+1](A) -> Repf (Gp)

again exact and fully faithful. We call pr(A) the essential image.

There is again a simple characterization of the category )pa(E) of E-representations of Gp lying in 0p" as a suitable full



11140 Colloquium Paper: Fontaine

subcategory of the category of semistable representations with crystalline semisimplification. Moreover:

If p + 2, the category of semistable Vvalues with hr(V) = 0 if r o {0, -1} is a full subcategory stable under extensions of Qp (E).

For 0 -< r <p -

1, let rr the full subcategory of Relpfz(Gp) of T's such that there is a filtration (necessarily unique)

0 = F,.+iT C FrT C ... F1T C FoT = T

such that griT(-i) is unramified for all i; then 2prd,r is a full

subcategory of pna stable under extensions.

Again, in ?5pa, deformations don't change Hodge type. The conductor may change.

Computation of Hlna(Qp, gt(Un)). As before, this can be translated in terms of the category MFN(CE) D

MFNh-P?10l(CE): if we define HMFN(Qp, M) as the ith-derived

functor, in the category MFN(CE), of the functor

HomMFN(%E)(CE, -), these groups are the cohomology of the

complex

FilM -- Fil-M M --> M -- 0- 0 -...

(with x 1-s (Nx, (1 -

?0)x) and (y, z) (-> (1 -

4-1)y -

Nz). Again, in this case, H&,,a(Qp, Qg(Un)) = ExtmrFN-P+1,0]()(Mn, Mn) = ExtMFN(A)(Mn, Mn) HmFN(Qp, Ende(M,,)). But,

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

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

Example 3: apt [the good generalization of ppr to the semistable case, theory due to Breuil (12)].

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

(a) FiliS as the ideal of S generated by the vm/m!, for m ' i;

(b) () as the unique Zp-endomorphism such that +(u) = uP;

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

For r < p -

1, (Pr: FilrS --> S is defined by (4r(x) = p-r()(x).

If r < p - 2, let 'A4o be the category whose objects consist of:

(i) an S-module A,

(ii) a sub-S-module FilrA of M containing FilrS.bt, (iii) a linear map (fr: Filr --> Jt, such that ()r(sx) = ()r(s).4)(x)

(where 0): X --> t is defined by +(x) = r(vrx)/4r(vr)), with an obvious definition of the morphisms. We consider the full

subcategory Jtr of 'tDo whose objects satisfy (i) as an S-module A -4 ?E-li-dS/pndS for suitable integers d

and (ni)l<i<d; (ii) as an S-module At is generated by the image of (fr. Finally, define tAr as the category whose objects are objects

Mt, of JE equipped with a linear endomorphism

N : --> iM

satisfying (i) N(sx) = N(s).x + s.N(x) for s E S, x Al, (ii) v.N(FilrJIU) C FilrFtM,

(iii) if x E Filrift, ) (v).N(fr(x)) = ()E(v.N(x)). This turns out to be an abelian Zp-linear category and we call

MFB-ro?](Zp) the opposite category. For A an CE-algebra, one can define in a natural way the

category MFB[-r?](A) (for instance, if A is artinian, an object of this category is just an object of MFB[-~E?(Zp) equipped with an homomorphism of A into the ring of the endomorphisms of this object).

Breuil defines natural "inclusions":

MFB[--<'?](A) C MFB[-E?](A) (ifr + 1 p - 2),

MF[-'?](A) C MFN[-r?](A) C MFB[-r'~(A).

Proc. Natl. Acad. Sci. USA 94 (1997)

Moreover, the simple objects of MF-ro?](k), MFNV-r,o (k), and

MFB[-r?ol(k) are the same. Breuil extends U to MFB[-r~?](A) and proves that this functor is again exact and fully faithful. We call pt'r(A) the essential image.

Let V be an E-representation of Gp. Breuil proves that, if V lies in 2t,r then Vis semistable and hm(V) = 0 if m>0 or m < -r. Conversely, it seems likely that if V satisfies these two conditions, Vlies in gst,r. This is true if r = 1, and it has been proven by Breuil if E = Qp and V is of dimension 2. More importantly, Breuil proved also

PROPOSITION (13). Let X be a proper and smooth variety over

Qp. Assume X as semistable reduction and let r, n G N with 0 < r < p-2; then Het(Xp, Z/pnZ) is an object of 2pt,r(Zp).

When working with ptr, deformation may change the Hodge type (the conductor also). The computation of Hst,r(Qp, gt(Un)) still reduces to a computation in MFB-ro] (E) (or equivalently in JAr). This computation becomes difficult in

general but can be done in specific examples. Final Remarks. Let L be a finite Galois extension of Qp

contained in Qp, CL the ring of integers and eL = eL/Ip.

(a) Call _L, the full subcategory of Repf z(Gp) whose objects are representations which, when restricted to Gal(Qp/Qp), extends to a finite and flat group scheme over CL. If eL

- p - 1, an

E-representation V lies in 2bp if and only if it becomes crystalline overL andhm(V) = 0 form {O0, -1}). IfeL <p - 1, Conrad

(14) defines an equivalence between p,L and a nice category of filtered modules equipped with a Frobenius and an action of

Gal(L/Qp). Using it, one can get the same kind of results as we described for 2p)r. For eL = p - 1, the same thing holds if we

require that the representation of Gal(Qp/Qp) extends to a connected finite and flat group scheme over CL.

(b) More generally, Breuil's construction should extend to

E-representations becoming semistable over L with hm(V) = 0 if m > 0 or < -(p - l)/eL (< -(p - i)/eL with a "grain de

sel"). (c) Let RepQp(Gp)crisL (resp. RepQp(p)st L) be the category

of Qp-representations V of Gp becoming crystalline over L

(resp. semistable) with hm(V) = 0 if m > 0 or m < -r. Let 2p"iris,L (resp. sptNrL) be the full subcategory of Repfz (Gp) consisting of T's for which one can find an object V of

RepQp(Gp)rcrisL (resp. RepQp(Gp)stL) Gp-stable lattices U' C U of Vsuch that T - U/U'. I feel unhappy not being able to prove the following:

Conjecture. CprisrL (resp. Cpt NrL): Let V be a Qp- representation of V lying in prisr (resp. 5p"jL). Then Van

object of RepQp(Gp)crisL (resp. RepQ (Gp)s). The only cases I know Cp'isL" are r = 0, r = 1, and eL '

p - 1, r -< p - 1, and eL = 1. The only cases I know Cpt, rL

are r = 0, r = 1, and eL < 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, Unite de Recherche Associ6e D0752.

1. Fontaine, J.-M. & Mazur, B. (1995) Geometric Galois Represen- tations, Hong Kong Conference on Elliptic Curves and Modular Forms (International Press, Boston), pp. 41-78.

2. P6riodes p-adiques, Ast6risque 223 (1994) Soc. Math. de France. 3. Faltings, G. (1989) in Algebraic Analysis, Geometry and Number

Theory (The Johns Hopkins University Press, Baltimore, MD), pp. 25-80.

4. Tsuji, T. On Syntomic Cohomology of Higher Degree of a Semi- Stable Family, preprint.

5. de Jong, A. J. (1995) Smoothness, Semi-Stability, and Alterations, preprint.

6. Wiles, A. (1995) Ann. Math. 141, 443-551. 7. Schlessinger, M. (1968) Trans. A. M. S. 130, 208-222. 8. Mazur, B. (1989) in Galois Groups Over Q (M.S.R.I. Publications,

Springer), pp. 385-437.



Colloquium Paper: Fontaine

9. Ramakrishna, R. (1993) Comp. Math. 87, 269-286. 10. Fontaine, J.-M. & Laffaille, G. (1982) Construction de Represen-

tations p-adiques Ann. Scient. E. N. S. 15, 547-608. 11. Fontaine, J.-M. & Messing, W. (1987) p-adic Periods and p-adic

etale Cohomology, Contemporary Mathematics 67, 179-207. 12. Breuil, C. (1995) Construction de Representationsp-adiques Semi-

Proc. Natl. Acad. Sci. USA 94 (1997) 11141

Stables, Universit6 de Paris-Sud, preprint. 13. Breuil, C. (1996) Cohomologie etale de Torsion et Cohomologie

Cristalline en Reduction Semi-Stable, Universit6 de Paris-Sud, preprint.

14. Conrad, B. (1996) Finite Honda Systems and Supersingular Elliptic Curves, Ph.D. (Princeton University, Princeton).



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Deforming Semistable Galois Representations