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Journal of Number Theory 147 (2015) 721748

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n some applications of integral p-adic Hodge eory to Galois representations

o Yamashita a,,1, Seidai Yasuda b,2

Toyota Central R&D Labs. Inc., 41-1, Yokomichi, Nagakute, Aichi, 480-1192, panDepartment of Mathematics, Graduate School of Science, Osaka University, saka, 560-0043, Japan

r t i c l e i n f o a b s t r a c t

rticle history:eceived 5 November 2012eceived in revised form 18 July 14ccepted 18 July 2014ailable online 5 October 2014ommunicated by D. Burns

eywords:utomorphy (modularity) lifting eoremstegral p-adic Hodge theoryach moduleoduli of Wach modules

We explicitly construct an analytic family of n-dimensional crystalline representations by using integral p-adic Hodge theory. This is a generalization of results by Berger, Li, and Zhu and by Dousmanis. We show, by using Kisins method, that the part of a universal deformation ring related to the above constructions is connected. From this we obtain an explicitly described subclass of potentially diagonalizable representations in the sense of Barnet-Lamb, Gee, Geraghty and Taylor. This yields automorphy lifting theorem and potential automorphy theorem, in which the condition at pis weakened.

2014 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/).

Corresponding author.E-mail addresses: [email protected], [email protected] (G. Yamashita),

[email protected] (S. Yasuda).Supported by 21st Century COE Program in Kyoto University Formation of an international center of

cellence in the frontiers of mathematics and fostering of researchers in future generations, EPSRC grant /E049109/1 in England, and TOYOTA Central R&D Labs., Inc.Partially supported by JSPS Grant-in-Aid for Scientic Research 21540013, 24540018.

tp://dx.doi.org/10.1016/j.jnt.2014.07.026

22-314X/ 2014 The Authors. Published by Elsevier Inc. This is an open access article under the CC Y license (http://creativecommons.org/licenses/by/3.0/).

722 G. Yamashita, S. Yasuda / Journal of Number Theory 147 (2015) 721748

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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7222. A family of Wach modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724

2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7242.2. Review of the theory of Wach modules and Wach lattices . . . . . . . . . . . . . . . . . . 7252.3. Some elementary properties of A+Qp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7282.4. The matrix P ((Xi,j,)i,j[1,n],) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7292.5. The matrix G(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7312.6. The matrix G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734

3. The representation of type (w, (ri,), (i,)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7383.1. Semi-induced representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

4. Local deformation rings and moduli of Wach modules . . . . . . . . . . . . . . . . . . . . . . . . . . 7435. Applications to automorphy lifting theorem and potential automorphy theorem . . . . . . . . 746cknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747eferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747

Introduction

Kisin introduced a new technique on automorphy (modularity) lifting theorems. The chnique is to study local deformation rings by using a moduli of semi-linear algebraic jects (S-modules, and Wach modules) in the integral p-adic Hodge theory, and to ow Rred = T by using the properties of local deformation rings. In the GL2 case, this ethod had an important application to Serres conjecture.On the other hand, Clozel, Harris and Taylor proved some automorphy lifting theo-ms for unitary groups in [CHT]. Their method has been improved in [G,T,BLGHT]n [T], Taylor used Kisins technique so-called in = p case) and has produced an portant application to SatoTate conjecture [HSBT,BLGHT]. However, the automor-

hy lifting theorem has not been established in the best possible form, in the sense that ey require certain local conditions of the involved Galois representations. A main dif-

culty in removing such local conditions lies in the study of local deformation rings in e case where the coecient eld has the same characteristic p as the residue eld of e considered local eld (so-called in = p case).Currently, one of the strongest available statements is given in [BLGGT], where an tomorphy lifting theorem has been established under some mild condition in the case here the Galois representation, restricted to the decomposition group at each prime ividing p, is potentially diagonalizable in the sense of [BLGGT]. However, the notion potential diagonalizability is dened in terms of a universal deformation ring whose ructure has not yet been well-understood. Owing to this, potential diagonalizability is some sense like a tautological synonym for the local condition with which we can man-e to show an automorphy of a global Galois representation by the currently available ethods, and it seems that we do not have a description of the potentially diagonalizable presentations which is concrete enough for applications.In this paper we construct, for any integer n 2 and for any nite unramied exten-on K of Qp, an explicitly described subclass of potentially diagonalizable n-dimensional

G. Yamashita, S. Yasuda / Journal of Number Theory 147 (2015) 721748 723

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pothadic representations (depending on a choice of some extra parameters w, r, which we troduce in Denition 2.7) of the absolute Galois group GK of K, by using the method Berger, Li and Zhu [BLZ], and Kisin [K1,K2] (see Proposition 5.2). More precisely,

First, we construct an explicit analytic family of Wach modules by the method of Berger, Li and Zhu (see Proposition 3.12 and Corollary 3.13).Next, we show a property on the connected components of local deformation rings by a method similar to that of Kisin (see Proposition 4.2).

he above extra parameters dene a p-adic representation of GK which is a direct sum representations induced from characters of open subgroups of GK corresponding to ite unramied extensions of K. The subclass consists of the representations which are adically near this direct sum. We call the members of the subclass the absolutely early semi-induced representations. From our construction, combined with the result [BLGGT], we deduce an automorphy lifting and a potentially automorphy result for essentially conjugate self-dual p-adic representation r of the absolute Galois group a CM eld F , such that F is unramied at p and for each prime v of F dividing p, e restriction of r to the decomposition group at v is absolutely nearly semi-induced a successive extension of one-dimensional representations (under some other mild nditions for r). See Corollary 5.3 and Corollary 5.4 for the precise statements. We phasize the following two things:

The notion of the potentially diagonalizability in [BLGGT] means the representations which are geometrically near to the induced representations. On the other hand, the notion of absolutely nearly semi-induced representations in this paper means the representations which are p-adically near to (the direct sums of) the induced representations.Our results are quantitative. We gave a p-adic estimate in which the automorphic lifting machinery works (see also Denition 3.11).

he extra parameter r stands for the HodgeTate weights and we can construct the sub-ass of representations starting from any xed HodgeTate weights (however, when we ply [BLGGT] after that, we have to assume that it has distinct HodgeTate weights). owever our results still remain partial and unfortunately they are not sucient to cover l the n-dimensional crystalline representations of GK with xed HodgeTate weights, cept for the case where the HodgeTate weights are in an interval of length p 1, d for some very limited case where n = 2 and the dierence of any two HodgeTate eights is in {p, 0, p}. It seems dicult3 to extend our method, even when n = 2 and e extra parameter w is not the identity, to cover the case where one of the eigenvalues

After a submission of this paper, we found a way of extending our method by using the hypergeometric

lynomials when n = 2 and K = Qp [YY]. We expect a possibility that this new technique is generalized to e case where n 2 and K is an unramied extension of Qp.


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of crystalline Frobenius is not close to zero. We note that, for such cases, a partial result as been obtained in [K2] when K = Qp and n = 2, based on the computation by Berger d Breuil of the reductions of some crystalline representations using the compatibil-

y of the mod p and the p-adic local Langlands correspondences for GL2(Qp) (see also emark 3.14).After a submission of our paper, a referee kindly informed us of the references [D,Z,

3,GL] whose subjects are related to that in our paper. All of these papers are concerned ith the computation of the reduction of a crystalline representation V of GK with K/Qpnramied. We briey explain their relations to our paper. The main results of [D] and e calculations of the reductions in [D] are exclusively for the case where V is two dimen-onal. However some two dimensional cases not covered in our paper are treated there. oreover in Section 4 of [D], Dousmanis gives a criterion for a pair (P, G) of matrices be an approximation of the pair of the matrices of and on some family of Wach odules. The approximation treats not only the case of the direct sum of the induced presentations like us, but also the case of more general ones. However he does not ex-licitly construct the family of Wach modules when the rank is greater than two. In this nse, our paper is a generalization of [D]. In [Z] and [B3] they proved, among some other sults, that any crystalline representation V in an explicitly given p-adic neighbourhood V has the same reduction as V . However they do not give an explicit computation the reduction. (They also treat other topics. However, they have little relation to our aper.) In [GL], they treat only the case where the length of the Hodge ltration is less an or equal to p 1. In this sense, our paper is a generalization of [GL] as well.Let us give a brief summary of the structure of the paper. In Section 2, we construct a rtain family of Galois representations by using the theory of Wach modules developed [B1]. Our family is a generalization of the family constructed in [BLZ]. In Section 3, e introduce the notion of nearly semi-induced representations. In Section 4, we recall e deformation rings introduced by Kisin [K2,K3] and check that the family of nearly mi-induced representations are contained in one connected component of the local eformation space. In Section 5, we describe some applications to automorphy lifting d potential automorphy.

A family of Wach modules

Let n 1 be a positive integer. In this section, we construct an analytic family n-dimensional crystalline representations by using the theory of Wach modules, and lculate its mod p reduction. This is a generalization of the work by Berger, Li and Zhu LZ].

1. Notation

In this section, we x a rational prime p. Let K be a nite unramied extension

Qp. Let val : K Z {} denote the normalized valuation on K, and let W K


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(Aofeqththove the valuation ring of K with respect to the valuation val. Let denote the Frobenius W , that is, the unique automorphism of the Zp-algebra W which induces the p-power

robenius automorphism on W/pW . Since K = W [1/p], the automorphism on W is iquely extended to the automorphism of K, which we denote by the same symbol by use of notation. Fix an algebraic closure K of K and put GK := Gal(K/K). We put

K := Gal(K(p)/K) and let : K Zp denote the p-adic cyclotomic character.

Let us consider the ring K[[]] of formal power series with coecients in K with spect to the formal variable . Let : K[[]] K[[]] denote the -semi-linear, -adically continuous endomorphism of the K-algebra K[[]] which sends to (1 +)p1. or K , let : K[[]] K[[]] denote the K-linear, -adically continuous endomor-ism of the K-algebra K[[]] which sends to

(1 + )() 1 =

n=1

n1i=0 (() i) n

n! .

his gives an action of the group K on the K-algebra K[[]].

2. Review of the theory of Wach modules and Wach lattices

Let A+K := W [[]] K[[]] denote the subring of formal power series with coecients W and put B+K := A

+K [1/p]. The rings A

+K and B

+K are subrings of K[[]] which are

able under the actions of and . We recall the denition of Wach modules and Wach ttices over A+K and over B

+K (cf. [B1,BLZ]).

Let

AK := W [[]][1/] ={ +

i=ai

i ai W, val(ai) as i

}

note the p-adic completion of the ring A+K [1/]. The ring AK is a discrete valuation ng and p is a uniformizer of AK . Let BK := AK [1/p] be the eld of fractions of AK . he endomorphism of A+K has a unique extension to a -semilinear endomorphism AK , which we also denote by , such that the latter is continuous with respect to e p-adic topology. The action of the group K on A+K has a unique extension to the tion of K on AK such that for each K , the automorphism : AK AK of K given by the action of is continuous with respect to the p-adic topology.Let R be a nite Zp-algebra. We extend the endomorphism and the action of K to K)R := RZp AK by R-linearity. A (, K)-module over (AK)R is a pair D = (D, D)

a nitely generated free (AK)R-module D and a -semi-linear endomorphism D of D, uipped with a continuous (with respect to the p-adic topology) semi-linear action of e group K such that for each K the semi-linear automorphism of D given by e action of commutes with the endomorphism D. We say that a (, K)-module D

er (AK)R is tale if the subset D(D) D generates D over (AK)R.


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toLwinwofpa p(DloLet E is a nite extension of Qp and let OE denote its ring of integers. We extend e endomorphism and the action of K to (BK)E := E Zp AK by E-linearity. (, K)-module over (BK)E is a pair D = (D, D) of a nitely generated free K)E-module D and a -semi-linear endomorphism D of D, equipped with a con-

nuous (with respect to the p-adic topology) semi-linear action of the group K such at for each K the semi-linear automorphism of D given by the action of com-utes with the endomorphism D. We say that a (, K)-module D over (BK)E is tale there exists an (AK)OE -lattice D0 D stable under the actions of and K such at D0 is an tale (, K)-module over (AK)OE .For a nite Zp-algebra R (resp. a nite extension R of Qp), a free R-representation of

K is a nitely generated free R-module T equipped with a continuous, R-linear action : GK EndR(T ) where we endow EndR(T ) with the p-adic topology. In [F], Fontaine nstructs a functor D from the category of free Zp-representations of GK to the category tale (, K)-modules over AK , and shows that the functor D gives an equivalence of e categories. By extending this functor by R-linearity, we have a functor, also denoted

y D, which gives an equivalence between the category of free R-representations of GK the category of tale (, K)-modules over (AK)R (resp. (BK)R).From now on we x a nite extension E of Qp and let OE denote its ring of integers.

et mE denote the maximal ideal of OE , and let F be the residue eld OE/mE . Below we se the subscripts ()E and ()OE which respectively mean the operations EQp() and E Zp (). Since W is unramied over Zp, the OE-algebra WOE = OE Zp W is equal to e direct product WOE =

W, where for each , the OE-algebra W is the ring

integers of a nite unramied extension of E, and the index set = E is a nite set hose cardinality c is equal to the greatest common divisor of [F : Fp] and [K : Qp]. The robenius automorphism = idOE of WOE induces a cyclic permutation : order c, and a -linear isomorphism W

= W() of WOE -algebras for each . or a WOE -algebra R and for , we put R = WWOE R so that R =

R. The

domorphism on (A+K)OE induces an isomorphism : (A+K)OE ,

= (A+K)OE ,()r each .In [B1] (see also [BLZ]), Berger gives a criterion for a free E-representation of GK

be crystalline which is described purely in terms of its associated (, K)-module. et us recall the statement of the criterion. We put q := ()/ =

pi=1

(pi

)i1 which

e regard an element in (B+K)E = (A+K [1/p])E . For an invertible element x = (x)

(BK)E =

(BK)E, and for a tuple a = (a) of integers indexed by , e let xa denote the element (xa ) in (BK)E . Let V be a free E-representation GK of dimension d := dimE V and let a = (a) and b = (b) be two tu-les of integers indexed by such that a b holds for every . Then free E-representation V of GK is crystalline with HodgeTate weights in the tu-

le [a, b] := ([a, b]) of intervals if and only if the associated (, K)-module (V ), D(V )) over (BK)E has a (B+K)E-submodule N(V ) D(V ) satisfying the fol-wing conditions:


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tuNthfo) The module N(V ) is free of rank d over (B+K)E .) The action of K on D(V ) preserves N(V ).) The action of K on N(V ) induces a trivial action on N(V )/N(V ).) We have D(V )(bN(V )) bN(V ) and the quotient bN(V )/D(V )(bN(V )) is

killed by qba. Here D(V )(bN(V )) denotes the (B+K)E-submodule of bN(V ) gen-

erated by the subset D(V )(bN(V )) bN(V ).

oreover, if V is a free E-representation of GK which is crystalline with HodgeTate eights in [a, b], then a (B+K)E-submodule N(V ) D(V ) satisfying the above conditions unique and the condition (4) holds even when we replace a, b with any tuples of integers = (a), b = (b) with a b for every such that the HodgeTate weights of are in the tuple [a, b] of intervals. Hence for a crystalline free E-representation V of

K , the notation N(V ) makes sense even if we do not specify a range of its HodgeTate eights.When V is crystalline, Berger [B1] also gives an isomorphism N(V )/N(V ) =

crys(V )E of ltered -modules with coecient in E. Here we endow N(V )/N(V )ith the structure of a ltered -module in the following way. The endomorphism of (V )/N(V ) is the one induced from the endomorphism (q/p)b of N(V ) for a su-ently large integer b. We lter N(V ) by putting FiliN(V ) := {x N(V ) | D(V )(x) N(V )} for i Z, and endow the E-vector space N(V )/N(V ) with the ltration duced by the ltration (FiliN(V ))iZ on N(V ).A Wach module over (B+K)E with HodgeTate weights in [a, b] is a free (B

+K)E-module

of nite rank equipped with a semi-linear action K AutE(N) of K which is ntinuous with respect to the p-adic topology of AutE(N) and with a -semi-linear ho-omorphism N : (B+K)E [1/] (B+K)E N (B

+K)E [1/()] (B+K)E N which commutes

ith the action of K such that the following conditions hold:

) The action of K on N induces a trivial action on N/N .) We have N (bN) bN and the quotient bN/N (bN) is killed by qba. Here

N (bN) denotes the (B+K)E-submodule of bN generated by the subset N (bN) bN .

(N, N ) is a Wach module over (B+K)E with HodgeTate weights in [a, b], then the tion of K on N (resp. the homomorphism N ) induces a semi-linear action of K the (BK)E-module D(N) = (BK)E (B+K)E N (resp. a -semi-linear endomorphism

D(N) of D(N)), and the (BK)E-modules together with the endomorphism D(N) and e action of K form an tale (, K)-module over (BK)E .Let V be a crystalline free E-representation of GK with HodgeTate weights in a ple [a, b] of intervals. If T is a GK-stable OE-lattice in V , then N(T ) := D(T ) (V ) is an (A+K)OE -lattice in N(V ). The map T N(T ) gives a bijection between e GK-stable OE-lattices in V and the (A+ )OE -submodules N of N(V ) satisfying the Kllowing conditions:


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C) The (A+K)OE -module N is free of rank d = dimE V .) The action of K on N(V ) preserves N .) We have D(V )(bN) bN and the quotient bN/D(V )(bN) is killed by qba.

Here D(V )(bN) denotes the (B+K)E-submodule of bN generated by the subset

D(V )(bN) bN .

e call an (A+K)OE -submodule N of N(V ) satisfying the above conditions a Wach lattice N(V ).

3. Some elementary properties of A+Qp

In this subsection, we assume that K = Qp. For later use we state the following two atements as lemmas. Proofs of the statements are easy and are left to the reader.

emma 2.1. For any Qp , the element ()/ is invertible in A+Qp . emma 2.2. The sequence {i(1 + )}i0 converges to 1 in A+Qp with respect to the , )-adic topology on A+Qp . emma 2.3. The sequence {i(q)}i0 converges to p in A+Qp with respect to the (p, )-adic pology.

roof. Since i(q) =p1

j=0 i((1 + )j), the claim follows from Lemma 2.2.

We dene a subring RQp of Qp[[]] as follows:

RQp :={

i0ci

i Qp[[]] val(ci) + i

p 1 0 for all i 0}

he subring RQp is stable under and the action of K . We endow the ring RQp with e linear topology such that the set

{piRQp +

(RQp jQp[[]]

) i, j Z0} ideals in RQp forms a fundamental system of neighbourhoods of 0. We easily see at the ring RQp is complete with respect to this topology and that the element q/p is vertible in RQp .Lemma 2.3 has the following corollary:

orollary 2.4. The sequence of invertible elements {i(q)/p}i0 converges to 1 in RQp .The following lemma can be veried by direct calculation:


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2.

inn

itO

Lw

deF(Aemma 2.5.

) The ring A+Qp is a closed subring in RQp .) The topology on A+Qp induced from that on RQp by the inclusion A

+Qp

RQp coincides the (p, )-adic topology on A+Qp .

By combining Corollary 2.4 with Lemma 2.1, we get the following:

orollary 2.6. For any Qp , the sequence {i(q/(q))}i0 converges to 1 in A+Qpith respect to the (p, )-adic topology.

For a map f : Z0 Z, we dene an element (f) RQp to be the innite product

(f) :=i0

(i(q)/p

)f(i).

follows from Corollary 2.4 that this innite product converges in RQp .It is straightforward to check the element (f) RQp has the following properties:

We have (f) RQp .For two maps f1, f2 : Z0 Z, we have (f1 + f2) = (f1)(f2).For a map f : Z0 Z, let Sf : Z0 Z denote the map

Sf(i) :={ 0 if i = 0,

f(i 1) if i 1.

Then we have ((f)) = (Sf).

4. The matrix P ((Xi,j,)i,j[1,n],)

Let us return to the case for a general nite unramied extension K of Qp. Fix an teger n 1 and put [1, n] := {1, . . . , n}. For a ring A, let Mn(A) denote the ring of -by-n matrices with entries in A. Let E be a nite extension of Qp and let OE denote s ring of integers. We x 0 and put h = h(0) for h = 1, . . . , c 1. We x an E-basis (e) of W0 , where the index set is a nite set of cardinality [K : Qp]/c. et (c) = (c,), denote the matrix of the OE-linear automorphism c of W0ith respect to the basis (e).We let (

A+K)OE

[[X]] :=(A+K

)OE

[[Xi,j,(1 i, j n, )

]]note the ring of formal power series with n2 [K : Qp]/c variables over (A+K)OE .

or , we put (A+ )OE ,[[X]] = (A+ )OE ,[[Xi,j,(1 i, j n, )]] so that K K+K)OE [[X]] =

(A

+K)OE ,[[X]].


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(renition 2.7. We dene a set DataK,E,n to be the set of triples (w, r, ) of an element w the n-th symmetric group Aut([1, n]), a tuple r = (ri,)(i,)[1,n] of integers ri, Zdexed by the pairs (i, ) [1, n] , and a tuple = (i,)(i,)[1,n] of elements i, W indexed by the pairs (i, ) [1, n] . We call an element in DataK,E,n a sic datum of degree n. We say that a basic datum (w, (ri,), (i,)) is eective if ri, non-negative for every (i, ) [1, n] . We say that a basic datum (w, (ri,), (i,))as distinct HodgeTate weights if the integers r1,, . . . , rn, are distinct for every .

Fix an eective basic datum (w, r, ) = (w, (ri,), (i,)) DataK,E,n of degree n. he goal of this subsection is to introduce a matrix P (X) Mn((A+K)OE [[X]]), which epends on the basic data (w, r, ).We put r,min = mini[1,n] ri, and r,max = maxi[1,n] ri,. Let w denote the auto-orphism of the set [1, n] which sends (i, ) [1, n] to (i, 1 ()) if = 0 and hich sends (i, 0) to (w(i), 1 (0)). Let us consider the following three conditions on e data w and (ri,, i,)(i,)[1,n] :

There exists a pair (i, ) [1, n] such that rwk(i,) = rk (),max holds for any k 0.There exists a pair (i, ) [1, n] such that rwk(i,) = rk (),min holds for any k 0.r,max r,min is independent of .

We dene to be := 1 if all of the conditions hold, and := 0 otherwise. For ample, if w has no xed points and if the integers (ri,)(i,)[1,n] are dierent with ch others, then neither of conditions holds and hence = 0 in this case. We put

r := + max

(r,max r,min). (2.1)

e need some more notation before introducing the matrix P (X). For and for (A+K)OE ,[[X]], let (f) (A+K)OE ,()[[X]] denote the formal power series

tained by applying on each coecient of f. For f = (f) (A+K)OE [[X]], t (f) denote the element in (A+K)OE [[X]] whose component at is equal to (f1 ()) for = 0 and whose component at 0 is equal to

(f1 (0))((

(c),Xi,j,

)i,j,

) (A+K)OE ,0 [[X]].

or f (A+K)OE ,[[X]] and for K , let (f) denote the formal power series ob-ined by applying on each coecient of f . For a matrix M in Mn(K[[]]) or in n((A+ )OE [[X]]), let (M) (resp. (M)) denote the matrix obtained by applying Kesp. ) on the entries of M .


spis

to

F

L[1fr

Er

unhoanan

w

ande

2.Let 1n be the identity n-by-n matrix, and let C be the permutation matrix corre-onding to w1, that is, C is the n-by-n matrix whose (i, j)-entry is w(i),j where ,

the Kroneckers delta.For each (i, ) [1, n] , let fi, : Z0 Z0 denote the map which sends j Z0

fi,(j) := rwj(i,).

or i, j [1, n] and for , we write

(fi,)/(fj,) =k0

z (k)i,j,

k.

et m denote the smallest non-negative integer such that pmz (k)i,j, Zp for every i, j , n], for every , and for every 0 k r1. Since (fi,)/(fj,) RQp , it follows om the denition of RQp that m (r 1)/(p 1).

xample 1. One of the simplest cases where we cannot apply FontaineLaaille theory is = p. In this case, if ri, = 0 or p for any i [1, n], , then we have m = 0. However, fortunately for n 3, this condition (ri, = 0, or p for any i [1, n], ) cannot ld with distinct HodgeTate weights. So, if r = p with distinct HodgeTate weights, d n 3, then we have m = 1. For n = 2, it can hold with distinct HodgeTate weights d we have m = 0 (see [BLZ, Remark 4.1.2, 1]).

We write z(k)i,j, = pmz (k)i,j, and put

zi,j, :=r1k=0

z(k)i,j,

k,

hich is, by the denition of m, an element in A+Qp .For , let Z denote the matrix in Mn(W[[]][[X]]) whose (i, j)-entry is zi,j, Xi,j,. For , dene a matrix P(X) Mn(W[[]][[X]]) to be

P(X) :={

(1n + Z0) diag(1,0qr1,0 , . . . , n,0qrn,0 ) C, if = 0,(1n + Z) diag(1,qr1, , . . . , n,qrn,), if = 0

d put P (X) := (P(X)) Mn((A+K)OE [[X]]). Here diag(1,qr1, , . . . , n,qrn,)notes the diagonal matrix whose diagonal entries are 1,qr1, , . . . , n,qrn, .

5. The matrix G(r)

For , we let G Mn(RQp) denote the matrixG := diag((f1,), . . . , (fn,)

).


It(a

L

fo

P

hfr

ItinL

Pd

fo

O follows from the properties described in Section 2.3 that the matrix G is invertible s a matrix with entries in RQp).

emma 2.8. We have

G = diag((q/p)r1, , . . . , (q/p)rn,

) (G1 ())r = 0 and

G0 = diag((q/p)r1,0 , . . . , (q/p)rn,0

)C(G1 (0))C

1.

roof. Let i [1, n]. We have fw(i,)(j 1) = fi,(j) for j 1. Hence the equality

(fi,) = (q/p)ri,((fw(i,))

)olds from which the claim immediately follows for = 0. The claim for = 0 follows om the equality

C(G1 (0))C1 = diag

(((fw(1),1 (0))

), . . . ,

((fw(n),1 (0))

)).

For and for K , we let G(r), denote the n-by-n matrix

G(G)1 = diag((f1,)/

((f1,)

), . . . , (fn,)/

((fn,)

)).

follows from Corollary 2.6 that the matrix G(r), is an invertible matrix with entries A+Qp . We put G

(r) = (G(r),) which we regard an element in Mn((A

+K)OE ) =

Mn((A+K)OE ,).

emma 2.9. We have

P (0)(G(r)

)= G(r)

(P (0)

).

roof. For a matrix M Mn((A+K)OE ) =

Mn((A+K)OE ,) and for , let M

enote the component of M at . By Lemma 2.8, we have the equalities

G(r), = diag

((q/(q)

)r1, , . . . , (q/(q))rn,) (G(r) )r = 0 and

G(r),0

= diag((

q/(q))r1,0 , . . . , (q/(q))rn,0 ) C(G(r) )0C1.

n the other hand, the equalitiesdiag((

q/(q))r1, , . . . , (q/(q))rn,) = (P (0))1P (0)


fo

fo

weqmdi

T

P

ar

P

wcoulisen

arr = 0 and

diag((

q/(q))r1,0 , . . . , (q/(q))rn,0 ) = C(P (0)0)1P (0)0C1

llow from the denition of P (0). By combining these four equalities, we have

G(r) = C(P (0)

)1P (0)C1 C(G(r) )C1

here C Mn(WOE ) =

Mn(W) denotes the matrix whose component at 0 is ual to C and whose other components are equal to the identity matrix. The three atrices C(P (0))1, P (0)C1, C(G(r) )C1 commute with each other since they are agonal matrices. Hence the right hand side of the above formula can be rewritten as

P (0)C1 C(G(r) )C1 C(P (0))1 = P (0)(G(r) )(P (0))1.hus we have G(r) = P (0)(G(r) )(P (0))1, which proves the claim. roposition 2.10. The entries of the matrix

P (X)(G(r)

)(P (X)

)1 G(r)e in r(A+K)OE [[X]].

roof. By the denition of P (X), we have

P (X)P (0)1 = 1n + Z, (2.2)

here Z Mn((A+K)OE [[X]]) =

Mn((A+K)OE ,[[X]]) denotes the matrix whose

mponent at is equal to Z for every . The matrix 1n+Z is congruent to 1n mod-o the maximal ideal of (A+K)OE [[X]]. Hence by (2.2), the matrix (P (X))(P (0))1 invertible in Mn((A+K)OE [[X]]). Thus to prove the claim, it suces to show that the tries of the matrix

Y :=(P (X)

(G(r)

)(P (X)

)1 G(r) ) (P (X))(P (0))1e in r(A+K)OE [[X]].By (2.2), we have

Y =(P (X)

(G(r)

) G(r) (P (X))) (P (0))1= P (X)

(G(r)

)(P (0)

)1 G(r) (1n + Z)

= (1n + Z)P (0)

(G(r)

)(P (0)

)1 G(r) (1n + Z).


W

Fh

mdfoW

li

2.

Psa

P

cemfo

hW

pW

toe also have P (0)(G(r) )(P (0))1 = G(r) , by using Lemma 2.9. Thus we have

Y = (1n + Z)G(r) G(r) (1n + Z)= ZG(r) G(r) (Z).

or h = 0, . . . , c 1, it follows from the denitions of G(r) and Z that the component at of the (i, j)-entry of Y is

zi,j,h(fj,h)/((fj,h)

) (zi,j,h)(fi,h)/((fi,h))ultiplied by

Xi,j,. Let Kh denote the eld of fractions of Wh . Then by the

enition of zi,j,h , the element zi,j,h pm(fi,h)/(fj,h) is in rKh [[]]. There-re, the component at h of the (i, j)-entry of Y is an element of rKh [[]] h [[]] = rWh [[]] multiplied by

Xi,j,. In particular, the (i, j)-entry of Y

es in r(A+K)OE [[X]]. We are done. 6. The matrix G

roposition 2.11. For each K , there exists a unique matrix G Mn((A+K)OE [[X]])tisfying the following conditions:

G G(r) mod r(A+K)OE [[X]], andP (X)(G) = G(P (X)).

Before going to the proof, we will introduce some notation. We put P0 :=(X) mod (A+K)OE [[X]] and Z0 := Z mod (A

+K)OE [[X]]. They are n-by-n matri-

s with entries in (A+K)OE/() = WOE [[X]]. For , we let D0, denote the diagonal atrix diag(1pr1, , . . . , nprn,) Mn(W) and put D0 = (D0,) Mn(WOE ). It llows from the denition of P0 that the equality

P0 = (1n + Z0)D0C (2.3)

olds, which shows that the matrix P0 is invertible as a matrix with entries in OE [[]][1/p].For and for f0, W[[X]], let (f0,) W()[[X]] denote the formal

ower series obtained by applying on each coecient of f0,. For f0 = (f0,) OE [[X]], let (f0) denote the element in WOE [[X]] whose component at is equal (f0,1 ()) for = 0 and whose component at 0 is equal to

(( (c) ) ) (f0,1 (0))

,Xi,j,i,j,

W0 [[X]].


Fen

(A

B

w

th

L

(1

(2

Pthdethofcothri

mor M0 Mn(WOE [[X]]), let (M0) denote the matrix obtained by applying on the tries of M0.Let M0 be an element in WOE [[X]] (resp. in Mn(WOE [[X]])). For any M +K)OE [[X]] (resp. for any M Mn((A+K)OE [[X]])) satisfying M0 M mod

(A+K)OE [[X]], the following properties are easily checked:

(M0) (M) mod (A+K)OE [[X]], and(M0) (M) mod (A+K)OE [[X]] for any K .

For M0 Mn(WOE [[X]]), put

(M0) := prP0(M0)P10 .

y (2.1) and (2.3), the component (M0) of (M0) at is equal to

(M0) = p(1n + Z0)(pr,minD0

)C(M0)C1

(pr,maxD10

)(1n + Z0)1, (2.4)

here = r (r,max r,min). Thus (M0) lies in Mn(WOE [[X]]), and any entries of (M0) are divisible by p if = 1. Let m be the Jacobson radical of WOE [[X]]. It is clear at the operation M0 (M0) is -semilinear. Hence induces a semilinear operation 0 := mod m on Mn(WOE [[X]]/m) = Mn(WOE OE F).The following is a key lemma for the proof of Proposition 2.11:

emma 2.12.

) The operation 0 is nilpotent, i.e., there exists an integer i 1 such that i0 = 0. Furthermore, we can take such an i satisfying i nc.

) The operation is m-adically nilpotent in the following sense: for any integer k 1, there exists an integer i, such that the entries of i(M0) are in mk for any M0 Mn(WOE [[X]]).

roof. First we prove (1). Suppose that = 1. Since 0 = 0 by (2.4), the claim is clear in is case. Let us assume = 0. For i, j [1, n] and for , let Ei,j, Mn(WOE [[X]])note the matrix such that the component at of the (i, j)-entry is equal to 1 and that e other components and other entries are equal to 0. Let denote the subset the elements satisfying = 0. Let I denote the subset of [1, n] which nsists of the elements (i, ) [1, n] satisfying ri, = r,min, and let J denote e subset of [1, n] which consists of the elements (i, ) [1, n] satisfying , = r,max.We consider the reductions modulo m of the both sides of (2.4). We have Z0 0od m, and (pr,minD0,) mod m is a diagonal matrix such that the component at

of the i-th diagonal entry is invertible if (i, ) I and is zero otherwise. On the other


hi-(2foan(wanwp

Shanan

T

Hfo

Ptrthk

wth

PM

Tand, (pr,maxD10,) mod m is a diagonal matrix such that the component at of the th diagonal entry is invertible if (i, ) J and is zero otherwise. Thus it follows from .4) that 0(Ei,j,) is a multiple of Ei,j,() for () = 0, and of Ew1(i),w1(j),0r () = 0 by a constant in W [[X]]/m, and 0(Ei,j,) = 0 unless w1(i, ) Id w1(j, ) J . (Note that w1(i, ) is equal to (i, ()) for () = 0, and to 1(i), 0) for () = 0.) Since we have assumed = 0, there exists, for any i, j [1, n]d for any , an integer k nc such that wk(i, ) / I or wk(j, ) / J . Hence e have nc0 (Ei,j,) = 0 for any i, j [1, n] and for any . Therefore nc0 = 0 which rove the claim (1).We prove the claim (2) by induction on k. If k = 1, the claim follows from (1).

uppose that k 2, and assume that the statement holds for k 1. By the induction ypothesis, there exists an integer i such that the entries of i0 (M0) are in mk1 for y M0 Mn(WOE [[X]]). Thus there exist an integer 0, elements x1, . . . , x m1, d N1, . . . , N Mn(WOE [[X]]) such that i

0 (M0) can be written as a nite sum

i

0 (M0) =

j=1xjNj .

hen, for i := i + nc, we obtain

i0(M0) =

j=1nc(xj)nc0 (Nj).

ence it follows from (1) that the entries of i0(M0) are in mk. Thus the statement holds r k. roof of Proposition 2.11. First we prove the uniqueness of G . Suppose that two ma-ices G and G , with G = G , satisfy the conditions of Proposition 2.11. Let k be e maximum non-negative integer satisfying G G mod k(A+K)OE [[X]]. We have r, since G G mod r(A+K)OE [[X]] by assumption. Put G G = kH, here H Mn((A+K)OE [[X]]). By assumption, we have P (X)(kH) = kH(P (X)), at is,

qkP (X)(H) = H(P (X)

). (2.5)

ut H0 := H mod (A+K)OE [[X]]. Then H0 is in Mn((A+K)OE [[X]]/()) =

n(WOE [[X]]), and H0 = 0 by the denition of k. By (2.5), we have pkP0(H0) = H0P0. hus we haveH0 = pkr(H0). (2.6)


B

fosiW

this

fo

siFT

P

Y

th

L

LanmTy applying (2.6) repeatedly, we have

H0 = pi(kr) i(H0)

r any i 1. It follows from Lemma 2.12 that the entries of the right hand de of the above formula converge to 0 with respect to the m-adic topology of OE [[X]]. Hence we have H0 = 0, which gives a contradiction. This proves that e matrix G satisfying the conditions of Proposition 2.11 is unique if it ex-ts.Next we will show the existence of G . It suces to construct Gk Mn((A+K)OE [[X]])

r each k r satisfying

G(k) G(r) mod r(A+K)OE [[X]],

G(k) G(k1) mod k1(A+K)OE [[X]] for k > r, and

P (X)(G(k) )(P (X))1 G(k) mod k(A+K)OE [[X]],

nce we can dene G as the limit of G(k) . We construct such a G(k) by induction on k. or k = r, we put G(k) := G(r) . We assume that k > r and that G(k1) is constructed. o construct G(k) , it is enough to nd a matrix H Mn((A+K)OE [[X]]) satisfying

P (X)(G(k1) + k1H

)(P (X)

)1 G(k1) + k1H mod k(A+K)OE [[X]].(2.7)

By assumption, there exists Y Mn((A+K)OE [[X]]) satisfying(X)(G(k1) )(P (X))1 G(k1) = k1Y . Put H1 := H mod (A+K)OE [[X]], 0 := Y mod (A+K)OE [[X]]. The equality (2.7) is equivalent to

H1 pk1P0(H1)P10 = Y0,

at is,

H1 pkr1(H1) = Y0. (2.8)

et us consider an innite sum

H1 :=i0

pi(kr1) i(Y0).

emma 2.12 shows that this innite sum m-adically converges, and satises (2.8). Take arbitrary lift H Mn((A+K)OE [[X]]) of H1. Then H satises (2.7). Therefore the atrix G(k) := G(k1) +k1H satises the conditions listed in the previous paragraph.

his completes the proof of the existence of G .


3.

F(A(fisac

rasLenP

reofratoa wis

Lti([la

N

([toT

Lfr The representation of type (w, (ri,), (i,))

We put

Tuple :=

i,j[1,n],mE .

or a = (ai,j,) Tuple, we dene an OE-algebra homomorphism eva : (A+K)OE [[X]] +K)OE to be the homomorphism which sends f(X) = (f(X)) (A+K)OE [[X]] to

h(ai,j,h(e)))0hc1 (A+K)OE . It is easy to check that the homomorphism eva compatible with the endomorphisms of (A+K)OE [[X]] and (A

+K)OE , and with the

tions of K on (A+K)OE [[X]] and (A+K)OE .

Let n 1 and let (w, r, ) DataK,E,n be an eective basic datum of degree n. We put ,min = mini[1,n] ri, and r,max = maxi,[1,n] ri, . Let P (X) and G for K be in the previous section. For a Tuple, we put Pa := eva(P (X)) and G,a := eva(G). et Na,OE = (A+K)

nOE

be a free (A+K)OE -module of rank n. Let N,a be the -semi-linear domorphism of Na,OE whose matrix with respect to the standard basis is equal to

a. For K , let a be the -semi-linear endomorphism of Na,OE whose matrix with spect to the standard basis is equal to G,a. The map a gives a semi-linear action the group K on Na,OE . We put Na = E OE Na,OE which is a free (B+K)E-module of nk n. We extend the endomorphism N,a of Na,OE (resp. the action of K on Na,OE ) that of Na (resp. that on Na) by E-linearity. The endomorphism N,a on Na induces -semi-linear homomorphism (B+K)E [1/] (B+K)E Na (B

+K)E [1/()] (B+K)E Na

hich we denote by the same symbol N,a by abuse of notation. The following lemma easily checked:

emma 3.1. The (B+K)E-module Na together with the endomorphism N,a and the ac-on of K form a Wach module over (B+K)E with the HodgeTate weights in the tuple r(),max, r(),min]) of intervals and the (A+K)OE -module Na,OE is a Wach ttice of the tale (, K)-module Na (B+K)E (BK)E over (BK)E. Let Va be the free E-representation of GK corresponding to the tale (, K)-module

a (B+K)E (BK)E over (BK)E . Then V is crystalline with the HodgeTate weights in r(),max, r(),min]) . Let Ta Va be the GK-stable OE-lattice corresponding the Wach lattice Na,OE . The following lemma follows from the argument in [BLZ, heorem 4.1.1].

emma 3.2. For a Tuple, there is a canonical isomorphism F OE Ta = F OE T0 of ee F-representation of GK where 0 = (0)i,j[1,n], Tuple.We need the following lemma.


Lsa

PanW

g(

C

is

C

isx

x

3.

Dtoa ch

nis

PD

ifEemma 3.3. Let and let s 1 be a positive integer. Let x be an element in (B+K)E,tisfying (x) qs(B+K)E,. Then x s(B+K)E,.

roof. Take a primitive p-th root p of unity in an algebraic closure K of K = EOEWd let W = W[p]. Let g be the E-linear (, p)-adically continuous automorphism of [[]] which sends to p(1 + ) 1.Since g((1 + )p) = (1 + )p, we have g = on W [[]]. Hence (x) = g((x)) q)sW [[]]. Since g(q) = q/(p(1 +) 1) is divisible by , we have (x) (B+K)E, sW [[]] = s(B+K)E,. This shows x s(B+K)E,. orollary 3.4. Let r, s 0 be two non-negative integers and let . Then the set

{x (B+K)E, qr(x) qs(B+K)E,}

equal to (B+K)E, if r s, and is equal to sr(B+K)E, if r < s. orollary 3.5. For each integer i 0, the set

FiliNa ={x Na

N,a(x) qiNa} equal to the set of the elements t((x1,), . . . , (xn,)) Na = (B+K)nE such that j,1 ()

irj,(B+K)E, for every (j, ) [1, n] with = 0 and i > rj,, and that w(j),1 (0)

irj,0 (B+K)E,0 for every j [1, n] with i > rj,0 . 1. Semi-induced representations

enition 3.6. We say that a free E-representation V is semi-induced if V is isomorphic a direct sum

i Vi of free E-representations of GK such that for each i, there exists

nite extension Ei of E such that Ei E Vi is induced from an Ei-valued continuous aracter of an open subgroup of GK .

Let L be a nite unramied extension of K and let OL denote its ring of integers. For , the OE Zp OL-algebra W W OL is equal to a direct product W W OL = W,, where for each , the OE-algebra W, is the ring of integers of a ite unramied extension of K, and the index set is a nite set whose cardinality

equal to the greatest common divisor of [F : Fp]/c and [L : K].

roposition 3.7. Let V be a crystalline free E-representation V of GK of dimension n. Let crys(V )E be the ltered -module over EQpK associated to V . Then V is semi-induced there exist a basic datum (w, r, ) = (w, (ri,), (i,)) DataK,E,n of degree n and an

Qp K-basis d1, . . . , dn of Dcrys(V )E which satisfy the following properties:


(1

(2

Mby

DE

th

Ran

Loneaa re

PV

E

exiscrsuT

Pto) For each i Z, the submodule FiliDcrys(V )E is equal to the set of elements nj=1 xjdj Dcrys(V )E such that the component of xj at 1 () is equal to zero

for every (j, ) [1, n] with = 0 and i > rj,, and that the component of xw(j)at 1 (0) is equal to zero for every j [1, n] with i > rj,0 .

) For , the component at of the matrix of with respect to the basis d1, . . . , dnis equal to

diag(1,p

r1, , . . . , n,prn,)

for = 0 and is equal to

diag(1,p

r1, , . . . , n,prn,) C

for = 0, where C denotes the permutation matrix corresponding to w1.

oreover the converse is true if V has distinct HodgeTate weights or if we replace V E E V for some nite extension E of E.

enition 3.8. For a basic datum (w, r, ) DataK,E,n and for a crystalline free -representation V of GK , we say that a V is semi-induced of type (w, r, ) if V satises e conditions in Proposition 3.7 for (w, r, ).

emark 3.9. We remark that, for a free E-representation V of GK which is crystalline d semi-induced, a basic data (w, r, ) as in Proposition 3.7 is not necessarily unique.

To prove Proposition 3.7 we need the following lemma.

emma 3.10. A free E-representation V of GK is semi-induced and crystalline if and ly if V is isomorphic to a direct sum

i Vi of free E-representations such that for

ch i there exist a nite extension Ei of E, a nite unramied extension Ki of K, and crystalline character i : GKi E i such that Ei E Vi is isomorphic to the induced presentation IndKKii.

roof. The if part of the claim is clear. We prove the only if part. Suppose that is semi-induced and crystalline. Then V is isomorphic to a direct sum

i Vi of free

-representations such that for each i, there exist a nite extension Ei of E and a nite tensions Ki of K and a continuous character i : GKi E i such that Ei E Vi

isomorphic to the induced representation IndKKii. Since V is crystalline, IndKKii is

ystalline for every i. This in particular shows that i is crystalline. Since the inertia bgroup of GK trivially acts on Dpst(IndKKii)Ei , the extension Ki/K is unramied. his proves the claim. roof of Proposition 3.7. Below we prove the converse part. The other part is easier

prove and follows by reversing the argument below.


a exiLE

thFfoexofx

Twhain

loTD

dmtaw

D

Dcr(wd1

(1

(2Suppose that V is crystalline and semi-induced. By Lemma 3.10, V is isomorphic to direct sum

i Vi of free E-representations such that for each i there exist a nite

tension Ei of E and a nite unramied extensions Ki and a crystalline character : GKi E i such that Ei E Vi is isomorphic to the induced representation IndKKii.

et us x i. By enlarging Ei if necessary, we may assume that the residual degree of i/Qp is divisible by [Ki : Qp]. Let rec : Ki GabKi denote the reciprocity map in e local class eld theory which sends a uniformizer in Ki to a lift of the geometric robenius, and let I(Ki) denote the set of embeddings Ki Ei of Qp-algebras. It llows from the classication of crystalline characters that for each I(Ki), there ists an integer r Z such that the character i rec is equal to the tensor product an unramied quasi-character i : Ki E i and the quasi-character which sends Ki to the product

I(Ki)(x)r E i .

he tensor product Ei Qp Ki is a product Ei Qp Ki =

I(Ki) Ei of copies of Ei

here x y Ei Qp Ki in the left hand side corresponds to ((x)y)I(Ki) in the right nd side. Let i denote the Frobenius automorphism of Ki. The composition with 1iduces a cyclic permutation of the set I(Ki) which we denote by I(Ki) : I(Ki) I(Ki).With the above notation, the ltered -module Dcrys(i)Ei over E

iQp Ki has the fol-

wing description. For I(Ki), let Dcrys(i) denote the component at of Dcrys(i)Ei . hen for j Z and for I(Ki), the component at of FiljDcrys(i)Ei is equal to crys(i) for j r, and is equal to zero for j > r. For each I(Ki), take an Ei-basis of Dcrys(i). Let I(Ki) and put = 1I(Ki)(). Let b Ei denote the unique ele-ent satisfying (d) = bd. Then we have

I(Ki) b = i(p). Since the character i

kes values in OEi , the element p bi(p) is in OEi . Hence by changing the basis d,

e may assume that b prOEi for each I(Ki).Then the claim follows from the observation that Dcrys(IndKKii)Ei is equal to

crys(i)Ei regarded as a ltered -module over Ei Qp K.

enition 3.11. For a basic datum (w, r = (ri,), = (i,)) DataK,E,n and for a ystalline free E-representation V of GK , we say that V is nearly semi-induced of type , r, ) if the ltered -module Dcrys(V )E over E Qp K associated to V has a basis , . . . , dn over E Qp K satisfying the following conditions:

) For each i Z, the submodule FiliDcrys(V )E is equal to the set of elements nj=1 xjdj Dcrys(V )E such that the component of xj at 1 () is equal to zero for

every (j, ) [1, n] with = 0 and i > rj,, and that the component of xw(j) at 1 (0) is equal to zero for every j [1, n] with i > rj,0 .

) Let m be the integer introduced in Section 2.4. (Note that we have m (r 1)/

(p 1).) Then for each there exists a matrix U GLn(W) which is congruent


Plety(B

L

Ctiof

Rthmcl

wsea locaTp-Hto the identity matrix modulo pmmEW such that the component at of the matrix of the endomorphism of Dcrys(V )E with respect to the basis d1, . . . , dn is equal to

U diag(1,p

r1, , . . . , n,prn,)

for = 0 and is equal to

U0 diag(1,p

r1, , . . . , n,prn,) C

for = 0.

The following Proposition is an immediate consequence of Corollary 3.5.

roposition 3.12. Let (w, r, ) DataK,E,n be an eective basic datum of degree n and t V be a free E-representation of GK which is crystalline and nearly semi-induced of pe (w, r, ). Then there exists a tuple a Tuple such that the Wach module N(V ) over +K)E is isomorphic to the Wach module Na. The following Corollary is an immediate consequence of the Proposition 3.12 and

emma 3.2.

orollary 3.13. Let V be as in Proposition 3.12. Then the semi-simplication of the reduc-on mod mE of any GK-stable OE-lattice of V is isomorphic to the semi-simplication F OE T0 as a free F-representation of GK.

emark 3.14. The condition U 1 (mod pmmEW) in Denition 3.11 implies that, for e representation V in Proposition 3.12 and Corollary 3.13, the characteristic polyno-ial ch(Frob) of the crystalline Frobenius Frob := [K:Qp] on Dcrys(V )E is p-adically ose to the characteristic polynomial ch(Qw,r,) of the matrix

Qw,r, := diag(

1,pr1, , . . . ,

n,prn,

)w1.

It is remarkable that for some two dimensional crystalline representations of GQp, for hich ch(Frob) are not p-adically close to ch(Qw,r,) for any (w, r, ) DataQp,E,2, the mi-simplication of its reduction mod mE is explicitly obtained in [BB] and [BG] by completely dierent method, based on the compatibility of the p-adic and the mod pcal Langlands correspondences. The latter method is eective for the other extreme se where ch(Frob) are nearly p-adically farthest from ch(Qw,r,) for any (w, r, ). he limitations n = 2 and K = Qp in the latter method come from the fact that the adic local Langlands correspondence has been established only under these conditions.

opefully they will be relaxed once the p-adic local Langlands correspondence will be


esfo

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Local deformation rings and moduli of Wach modules

In this section, we study universal family of Wach modules, local deformation rings, d subfamily corresponding to the family constructed in the previous section.Let AROE denote the category of Artinian local OE-algebra with residue eld F. Let VF

e a free F-representation of GK of dimension n. We consider the functor DVF from AROEe category of groupoids which sends an object A in AROE to the groupoid DVF(A) of formations of VF to A, that is, the groupoid DVF(A) such that the objects in DVF are e pairs of a free A-representation VA of GK and an isomorphism VAAF = VF of free -representations of GK , and for two objects VA,1 and VA,2 of DVF(A), the morphisms om VA,2 to VA,2 are the isomorphisms VA,1

= VA,2 of free A-representations of GKhich are compatible with the given isomorphisms VA,i A F = VF for i = 1, 2. Fix an -basis of VF. We also consider the functor DVF from AROE to the category of sets which nds an object A in AROE to the set DVF(A) of framed deformations (or, liftings in e terminology of [CHT]) of VF to A, that is, the set DVF(A) of isomorphism classes triples of a free A-representation VA of GK , an isomorphism VA A F = VF of free -representations of GK , and a lifting to VA of the xed basis of VF. Here two such iples VA,1 and VA,2 are called isomorphic if there exists an isomorphism VA,1

= VA,2hich transports the given basis of VA,1 to the given basis of VA,2. The functor DVF is o-representable by a complete local Noetherian OE-algebra RVF .Take an object A in AROE , and let VA be an object of the groupoid DVF(A). Let A := D(V A) be the tale (, K)-module over (AK)A associated to the A-linear dual A of VA. For a tuple r = (r) of non-negative integers indexed by , let us consider e functor WVA,r from the category of A-algebras to the category of sets, which sends A-algebra B to the set of Wach lattices in MA A B with HodgeTate weights in the ple [0, r] of intervals. (see [K2, 3] for the denition of Wach lattices in MA A B). y [K2, Proposition (3.5)] the functor WVA,r has the following properties: The functor VA,r is represented by a projective A-scheme WVA,r. If R is any complete local oetherian ring with residue eld F, and VR a deformation of VF to R, then there exists projective R-scheme R : WVR,r SpecR such that

) If A is in AROE and R A is a homomorphism of local OE-algebras inducing the identity on residue elds, then there exists a canonical isomorphism WVR,rRA

=WVA,r, where VA = VR R A.

After a submission of this paper, we found a way of extending our method by using the hypergeometric lynomials when n = 2, K = Qp and ch(Frob) are between p-adically close to ch(Qw,r,) and nearly adically farthest from ch(Qw,r,), which covers all the intermediate range if the width of Hodge ltration

p2+1less than 2 [YY]. We also expect a possibility that this new technique is generalized to the case where 2 and K is an unramied extension of Qp.


(2

cr

anar

(1(2

Lex

(1

(2

P

ouWF

tha b) The map R : WVR,r SpecR becomes a closed embedding after inverting p. If Ais a nite E-algebra, then a homomorphism R A of OE-algebras factors through the coordinate ring of the scheme theoretic image of R if and only if VA = VR R Ais a crystalline representation with HodgeTate weights in [0, r].

For R = RVF , let R,rVF denote the coordinate ring of the scheme theoretic image of RVF

. For a nite local Qp-algebra A and a free A-representation VA of GK which is ystalline with non-negative HodgeTate weights, we put

PVA(T ) := detKQpA(T [K:Qp]Dcrys(V A)),

d let cVA,i denote the coecient of T i in PVA(T ) for i = 0, . . . , n 1. It follows from the gument in [K2, Corollary (3.7)] that the OE-algebra R,rVF has the following properties.

) R,rVF [1/p] is formally smooth over E.) For i = 0, . . . , n 1, there is a unique ci R,rVF [1/p] such that for any nite E-algebra A, and any OE-algebra homomorphism h : RVF A, the representation VA obtained from the universal representation over RVF by specialization by h sat-ises cVA,i = h(ci). Moreover, the element ci is contained in the normalization of Im{R,rVF R,rVF [1/p]}.

emma 4.1. For any nc-tuple r = (ri,)(i,)[1,n] of integers and for every , there ists a unique quotient R,rVF of RVF with the following properties:

) R,rVF is p-torsion free, and if R,rVF = 0, then R,rVF [1/p] is formally smooth over Eof pure dimension n2 + ([K : Qp]/c)

m where for , let m denote the

number of pairs (i, j) [1, n] [1, n] satisfying ri, > rj,.) If E is a nite extension of E, x : RVF E a morphism of E-algebras, and Vx the

representation of GK obtained by specializing the universal RVF-representation by x, then x factors through R,rVF if and only if Vx is crystalline with HodgeTate weights (r1,z(), . . . , rn,z())E where z : E = E denotes the map induced by the inclusion WOE WOE .

roof. This is a special case of [K3, Theorem (3.3.8)]. Fix an eective basic datum (w, r, ) = (w, (ri,), (i,)) DataK,E,n. We restrict rselves to the case where VF is the F-linear dual of F OET0 where T0 is as in Lemma 3.2. e x an F-basis of VF. Let a Tuple. By Lemma 3.2, we have a canonical isomorphism

OE T a = VF. Let B be an OE-basis of the OE-dual T a of Ta which is compatible with e xed basis of VF under the above isomorphism F OE T a = VF. The pair (Ta, B) gives homomorphism R,r OE of local OE-algebras. Let xa,B : R,r[1/p] E denote its VF VFase change to E.


PE

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Siinroposition 4.2. The connected component of SpecR,rVF [1/p] which contains the-rational point xa,B does not depend on the choice of the pair (a, B).

roof. First we prove that the connected component does not depend on the choice B. Let (a, B1) and (a, B2) be two pairs with the common rst component. Then ere exists an element g in the kernel of GLn(OE) GLn(F) such that B2 = gB1, at is, the matrix of the identity endomorphism of T a with respect to the basis B2 the source and the basis B1 in the target is equal to g. We x a uniformizer of and write g = 1n + Y . Let R := OE [[t]] be the ring of formal power series with ecients in OE . For an object A in AR(OE) and for a homomorphism f : R A local OE-algebras, the pair (A OE Ta, (1 + f(t)Y )B1) denes an element in DVF(A). he association f (A OE Ta, (1 + f(t)Y )B1) yields a homomorphism : RVF R complete local OE-algebras. We claim that the homomorphism factors through e quotient R,rVF and hence induces a homomorphism : R,rVF R. For a m, t fa : R E denote the continuous homomorphism of OE-algebras which sends to a. It follows from the construction of the homomorphism that the composite factors through the quotient R,rVF for any a m. Then the claim follows since eierstrass preparation theorem implies that the homomorphism R amE given fa is injective. It is clear that the homomorphism xa,B1 (resp. xa,B2) is equal to the mposite

R,rVF [1/p] R[1/p] E

here the second map is the continuous homomorphism which sends t to 0 (resp. 1). nce SpecR[1/p] is irreducible, the two points xa,B1 and xa,B2 lie in the same irreducible mponent of SpecR,rVF [1/p].Let R be a quotient of the ring OE [[X]] = OE [[Xi,j,(i, j [1, n], )]] such at R is a nite OE-algebra. Let sR : OE [[X]] R denote the canonical surjection. et P (X)R and G,R denote the matrices obtained by applying the homomorphism sR : (A+K)OE [[X]] = OE [[X]] Zp A+K R Zp A+K to the entries of P (X) and Gspectively. Then the pair (P (X)R, (G,R) ) gives a structure of tale (, K)-module (RZp AK)n. Let TR denote the free R-representation of GK associated to this tale , K)-module. Let T R denote the R-linear dual of TR. The collection (T R)R, where Rns over the quotients of the ring OE[[X]] which is nite over OE , forms a projective stem. We put T OE [[X]] = limR T

R and take an OE [[X]]-basis BOE [[X]] which lifts the

ven basis of T F = VF. The pair (T OE [[X]], BOE [[X]]) gives a framed deformation of F to OE [[X]]. We put r,max = maxi[1,n] ri, for and rmax = (r,max) . follows from the construction of the OE[[X]]-scheme W = WTOE [[X]],rmax that the atrices P (X) and G give an OE [[X]]-rational point of W . Hence the homomorphism VF

OE [[X]] which comes from the universality factors through the quotient R,rmaxVF . nce the set of E-rational points of a rigid analytic open unit disc over E is not contained

any proper analytic closed subvariety, a technique similar to that in the proof of the


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(5aim in the previous paragraph shows that the homomorphism RVF OE [[X]] factors rough the quotient R,rVF . Since the scheme SpecOE [[X]][1/p] is irreducible, the claim llows. Applications to automorphy lifting theorem and potential automorphy theorem

enition 5.1. We say that a crystalline free E-representation V of GK is absolutely early semi-induced if there exist a nite extension E of E and a basic datum (w, r, ) =, (ri,), (i,)) DataK,E,n such that E E V is nearly semi-induced of type (w, r, ). e say that a free E-representation V of GK is upper-triangular if there is a ltration V by E-subrepresentations whose graded pieces are one-dimensional.

Proposition 4.2 together with Lemma 1.4.1 and Lemma 1.4.3 of [BLGGT] gives the llowing result.

roposition 5.2. Let V be a free E-representation of GK which is either crystalline d absolutely nearly semi-induced or potentially crystalline and upper-triangular. Then E OQp is potentially diagonalizable in the sense of [BLGGT, 1.4]. This proposition, together with the main results of [BLGGT, Theorem 4.2.1, Corollary

5.2], we have the following automorphy lifting and potential automorphy results.

orollary 5.3. Let F be an imaginary CM eld and let F+ denote its maximal totally al subeld. Let n 1 be an integer and let p be a prime number such that F is nramied at p, and that F does not contain a primitive p-th root of unity. Fix an omorphism : Qp

= C. Let E be a nite extension of Qp and let (r, ) be the pair of continuous absolutely irreducible representation r : GF GLn(E) and a continuous aracter : GF+ E such that rc is isomorphic to r where c Gal(F/F+)notes the complex conjugation. Let r denote the reduction of r and assume that the l lowing conditions are satised:

) r is unramied outside a nite set of primes.) The restriction of r to GF (p) is irreducible.) p 2(d + 1) where d is the maximal dimension of an irreducible subrepresentation

of the restriction of r to the closed subgroup of GF generated by all Sylow pro-psubgroups.

) For all prime v of F dividing p, the restriction r|GFv is potentially crystalline with distinct HodgeTate weights and is either crystalline and absolutely nearly semi-induced or upper-triangular.

) There is a RAECSDC automorphic representation (, ) of GLn(AF ) whose asso-ciated pair (rp,(), rp,()) of a p-adic Galois representation rp,() and a p-adic

character rp,() satises the following conditions:


T

Cnofr

isof

(1(2(3

T(r

A

thmthveGA

pthitofph

R

[B (r, ) is isomorphic to (rp,(), rp,()). either is -ordinary in the sense of [G, Denition 5.1.2.], or for every v|p, the

restriction rp,()|GFv is either crystalline and absolutely nearly semi-induced or potentially crystalline and upper-triangular.

hen (r, ) is automorphic of level potentially prime to p.

orollary 5.4. Let F+ be a totally real eld. Let n 1 be an integer. Let p be a prime umber such that p 2(n + 1) and p is unramied in F+. Let E be a nite extension Qp and let (r, ) be the pair of a continuous absolutely irreducible representation : GF+ GLn(E) and a continuous character : GF+ E such that the pair (r, ) a totally odd, essentially conjugate self dual representation. Let r denote the reduction r and assume that the following conditions are satised:

) r is unramied outside a nite set of primes.) The restriction of r to GF+(p) is irreducible.) For all prime v of F+ dividing p, the restriction r|G

F+v

is potentially crystalline with distinct HodgeTate weights and is either crystalline and absolutely nearly semi-induced or upper-triangular.

hen there exists a Galois, totally real extension F+/F+ such that restriction |G

F+ , |GF+ ) is automorphic of level prime to p.

cknowledgments

This paper is a generalized version of the preprint (2008) of the rst author. He thanks e second author for discussions and for extending the construction of the family of Wach odules in the previous version, and for checking and correcting the other part. (He also anks him for precisely citing the recent preprints to remove some parts of the previous rsion. Especially, by citing [BLGGT], we could remove the part showing Rred = T for Ln case.) He thanks Ivan Fesenko in Nottingham University for the hospitality from pril/2008 to March/2010.He sincerely thanks TOYOTA Central R&D Labs., Inc. for oering him a special

osition in which he can concentrate on pure mathematical research. The president of e company at that time told him that if they forced him to do something else, then

would be against the policy of the company, and it would be against the philosophy the founder of the company as well. He also sincerely thanks Sakichi Toyoda for his ilosophy, and the executives for inheriting it from him for 80 years after his death.

eferencesLGGT] T. Barnet-Lamb, T. Gee, D. Geraghty, R. Taylor, Potential automorphy and change of weight, Ann. of Math. (2) 179 (2014) 501609.


[BLGHT] T. Barnet-Lamb, D. Geraghty, M. Harris, R. Taylor, A family of CalabiYau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (1) (2011) 2998.

[B1] L. Berger, Limites de reprsentations cristallines, Compos. Math. 140 (2004) 14731498.[B2] L. Berger, Reprsentations modulaires de GL2(Qp) et reprsentations galoisiennes de dimen-

sion 2, Astrisque 330 (2010) 263279.[B3] L. Berger, Local constancy for the reduction mod p of 2-dimensional crystalline representa-

tions, Bull. Lond. Math. Soc. 44 (3) (2012) 451459.[BB] L. Berger, C. Breuil, Sur la rduction des reprsentations cristallines de dimension 2 en poid

moyens, unpublished note, which is contained in [B2].[BLZ] L. Berger, H. Li, H.J. Zhu, Construction of some families of 2-dimensional crystalline repre-

sentations, Math. Ann. 329 (2) (2004) 365377.[BG] K. Buzzard, T. Gee, Explicit reduction modulo p of certain two-dimensional crystalline rep-

resentations, Int. Math. Res. Not. IMRN (12) (2009) 23032317.[CHT] L. Clozel, M. Harris, R. Taylor, Automorphy for some -adic lifts of automorphic mod Galois

representations, Publ. Math. Inst. Hautes Etudes Sci. 108 (2008) 1181.[D] G. Dousmanis, On reductions of families of crystalline Galois representations, Doc. Math. 15

(2010) 873938.[F] J.-M. Fontaine, Reprsentations p-adiques des corps locaux I. The Grothendieck Festschrift,

vol. II, Progr. Math., vol. 87, Birkhuser Boston, Boston, MA, 1990, pp. 249309.[GL] H. Gao, T. Liu, A note on potential diagonalizability of crystalline representations, preprint,

[G

[H

[K

[K[K

[T

[Y

[Zhttp://arxiv.org/abs/1204.4704.] D. Geraghty, Modularity lifting theorems for ordinary Galois representations, PhD thesis,

Harvard University, ISBN 978-1124-07900-4, 2010, 131 pp., ProQuest LLC.SBT] M. Harris, N. Shepherd-Barron, R. Taylor, A family of CalabiYau varieties and potential

automorphy, Ann. of Math. 171 (2010) 779813.2] M. Kisin, Modularity for some geometric Galois representations, in: L-Functions and Galois

Representations, Durham, 2004, pp. 438470.3] M. Kisin, Potentially semistable deformation rings, J. Amer. Math. Soc. 103 (2008) 513546.1] M. Kisin, Moduli of nite at group schemes and modularity, Ann. of Math. 170 (2009)

10851180.] R. Taylor, Automorphy for some -adic lifts of automorphic mod Galois representations II,

Publ. Math. Inst. Hautes Etudes Sci. 108 (2008) 183239.Y] G. Yamashita, S. Yasuda, Reduction of two dimensional crystalline representations and Hy-

pergeometric polynomials, in preparation.] H.J. Zhu, Crystalline representations of GQpa with coecients, preprint, http://arxiv.org/

abs/0807.1078.

On some applications of integral p-adic Hodge theory to Galois representations1 Introduction2 A family of Wach modules2.1 Notation2.2 Review of the theory of Wach modules and Wach lattices2.3 Some elementary properties of AQp+2.4 The matrix P((Xi,j,)i,j [1,n],)2.5 The matrix G(r)2.6 The matrix G

3 The representation of type (w,(ri,),(i,))3.1 Semi-induced representations

4 Local deformation rings and moduli of Wach modules5 Applications to automorphy lifting theorem and potential automorphy theoremAcknowledgmentsReferences

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