Torsion <span class="mathjax-formula">$p$</span>-adic Galois representations and a conjecture of Fontaine

TORSION p -ADIC GALOIS REPRESENTATIONS AND A CONJECTURE OF FONTAINE

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ABSTRACT. – Let p be a prime, K a finite extension ofQp and T a finite free Zp-representation of Gal( ¯K/K). We prove thatT ⊗ZpQpis semi-stable (resp. crystalline) with Hodge–Tate weights in {0, . . . , r}if and only if, for alln,T /pⁿTis torsion semi-stable (resp. crystalline) with Hodge–Tate weights in{0, . . . , r}.

©2007 Elsevier Masson SAS

RÉSUMÉ. – Soientpun nombre premier,r un entier positif,K une extension finie deQp etT une Zp-représentation deGal( ¯K/K)libre de rang fini en tant queZp-module. On montre queT⊗ZpQpest semi-stable (resp. cristalline) à poids de Hodge–Tate dans{0, . . . , r}si et seulement si, pour tout entiern, la représentationT /pⁿTest le quotient de deux réseaux dans une représentation semi-stable (resp. cristalline) à poids de Hodge–Tate dans{0, . . . , r}.

©2007 Elsevier Masson SAS

1. Introduction

Let k be a perfect field of characteristicp,W(k) its ring of Witt vectors,K0=W(k)[¹_p], K/K₀a finite totally ramified extension ande=e(K/K₀)the absolute ramification index. For many technical reasons, we are interested in understanding the universal deformation ring of a fixed residual representation of G:= Gal( ¯K/K). In particular, it is important to study those deformations that are semi-stable (resp. crystalline). In [12], Fontaine conjectured that there exists a quotient of the universal deformation ring parameterizing semi-stable (resp. crystalline) representations. To prove the conjecture, it suffices to prove the following:

CONJECTURE 1.0.1 [12]. – Fix an integerr >0. Let T be a finite freeZp-representation ofG. ThenT⊗ZpQpis semi-stable(resp. crystalline)with Hodge–Tate weights in{0, . . . , r}if and only if, for alln,T_n:=T /pⁿT is torsion semi-stable(resp. torsion crystalline)with Hodge–

Tate weights in{0, . . . , r}, in the sense that there existG-stableZp-latticesL_(n)⊂L_(n)inside a semi-stable(resp. crystalline)Galois representationV_(n)with Hodge–Tate weights in{0, . . . , r} such thatT_nL_(n)/L_(n)asZp[G]-modules.

If T /pⁿT comes from the generic fiber of a finite flat group scheme over OK, i.e., in the case thatr= 1andV_(n)is crystalline for alln, the conjecture has been proved by Ramakrishna ([21]). The case that e= 1andV_(n) is crystalline has been proved by L. Berger ([2]), and the case thate= 1andr < p−1was shown by Breuil ([6]). In this paper, we give a complete proof of Conjecture 1.0.1 without any restriction. Our main input is from [14], where Kisin proved that anyG-stableZp-lattice in a semi-stable Galois representation is of finiteE(u)-height. More

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

precisely, fix a uniformiserπ∈Kwith Eisenstein polynomialE(u). LetK_∞=

n1K(^pn√ π), G_∞= Gal( ¯K/K_∞)andS=W(k)JuK. We equipSwith the endomorphismϕwhich acts via Frobenius on W(k), and sends uto u^p. For every positive integer r, let Mod^r,fr_/S denote the category of finite freeS-modulesMequipped with aϕ-semi-linear mapϕ:M→Msuch that the cokernel ofϕ^∗= 1⊗ϕ:S⊗ϕ,SM→Mis killed byE(u)^r. Such modules withϕ-structure are calledϕ-modules of finiteE(u)-height. For anyM∈Mod^r,fr_/S, one associates a finite free Zp-representationTS(M) of G_∞ ([8]). Kisin ([14]) proved that anyG_∞-stableZp-lattice L in a semi-stable Galois representation arises from aϕ-module of finiteE(u)-height, i.e., there existsL∈Mod^r,fr_/Ssuch thatT_S(L)L. In particular, this result implies that if aZp[G]-module M is torsion semi-stable with Hodge–Tate weights in {0, . . . , r}then there exists a (p-power) torsion ϕ-module M of height r (see §2 for precise definitions) such that TS(M)M as G_∞-modules. Therefore, we can use torsion ϕ-modules of finiteE(u)-height to study torsion representations ofG_∞. Ifp >2 andr= 1, Breuil and Kisin proved that there exists an anti- equivalence between the category of finite flat group schemes overOK and torsionϕ-modules of height1 ([14], [4]). Thus, torsionϕ-modules of finiteE(u)-height can be seen as a natural extension of finite flat group schemes overOK. In particular, we extend many results on finite flat group schemes overOKto torsionϕ-modules of finiteE(u)-height. For example, under the hypotheses of Conjecture 1.0.1, we prove that theZp-representationT in Conjecture 1.0.1 must arise from aϕ-module of finiteE(u)-height, i.e., there existsM∈Mod^r,fr_/Ssuch thatTS(M)T asG_∞-modules. To prove this result, we extend Tate’s isogeny theorem onp-divisible groups to finite level as in [19] and [3], i.e., we show that the functorTS is “weakly” fully faithful on torsion objects. (See Theorem 2.4.2 for details.)

So far, only theG_∞-action onThas been used. To fully use theG-action onT, we construct anA_cris-linear injection (in §5)

ι:M⊗S,ϕA_cris→T^∨⊗ZpA_cris (1.0.1)

such thatι is compatible with Frobenius and G_∞-action (cf. Lemma 5.3.4). Note that T is a representation of G. There is a natural G-action on the right-hand side of (1.0.1). However, it is not clear if M⊗S,ϕAcris is G-stable (viewed as a submodule of T^∨⊗ZpAcris via ι).

In §6 we prove thatM⊗S,ϕB_cris⁺ is stable under theG-action after very carefully analyzing

“G-action” onM/pⁿM⊗S,ϕAcris for eachn. In fact, we show that G(M) lies inM⊗S,ϕ

RK0 for a subring RK0 of B_cris⁺ . Finally, we prove that RK0 is small enough to show that dimK0(T^∨⊗_ZpB_st⁺)^Grank_Zp(T)and thus prove Conjecture 1.0.1. Let us apply our theorem to the universal deformation ring of Galois representations. LetE/Qpbe a finite extension with finite residue fieldF. Denote byᏯthe category of local Noetherian completeOE-algebras with residue fieldF. ForA∈Ꮿ, anA-representationT ofGis anA-module of finite type equipped with a linear and continuous action ofG. Fix a finite freeF-representationρ¯which is torsion semi-stable (resp. crystalline) with Hodge–Tate weights in{0, . . . , r}. LetD(A)be the set of isomorphism classes of finite freeA-representations T such thatT /m_OET ρ¯andD^ss,r(A) (resp.D^cris,r(A)) the subset ofD(A)consisting of isomorphism classes of those representations that are torsion semi-stable (resp. crystalline) with Hodge–Tate weights in{0, . . . , r}. By [20]

and [21], if H⁰(G,GL( ¯ρ)) =F, then D(A), D^ss,r(A) and D^cris,r(A) are pro-representable by complete local Noetherian rings Rρ¯, R^ss,r_ρ¯ andR^cris,r_ρ¯ , respectively. R^ss,r_ρ¯ andR^cris,r_ρ¯ are quotients ofRρ¯.

THEOREM 1.0.2. –For any finite K0-algebra B, a map x:Rρ¯→B factors through R^ss,r_ρ¯ (resp. R^cris,r_ρ¯ ) if and only if the induced B-representation Vx of G is semi-stable (resp.

crystalline)with Hodge–Tate weights in{0, . . . , r}.

e ◦

In fact, the existence of such a quotient ofRρ¯satisfying the property in the above theorem has been known by Kisin (cf., Theorem in [16]). Here we reprove the Theorem in [16] and further show that such quotient is justR^ss,r_ρ¯ (orR^cris,r_ρ¯ ). As explained in the introduction of [16], it will be useful to distinguish four flavors of the statement that some propertyP(e.g., being crystalline, semi-stable etc.) ofp-adic Galois representations cuts out a closed subspace of the generic fiber ofSpecRρ¯.

(1) Let E/Qp be a finite extension and xi:Rρ¯ → E (i 1) a sequence of points converging p-adically to a point x:Rρ¯→E. Write Vxi andVx for the corresponding E-representations. If theVxihaveP, thenVxhasP.

(2) The set {x∈HomE(Rρ¯,Cp)|xhasP}cuts out a closed analytic subspace in the rigid analytic space associated toRρ¯(see [1] for the more precise statement).

(3) There is a quotientR^P_ρ¯ of R_ρ¯such thatR_ρ¯→E factors through R^P_ρ¯ if and only ifV_x hasP.

(4) LetV be a finite dimensionalE-representation ofG, andL⊂V aG-stableZp-lattice.

Suppose that for eachn,L/pⁿLis a subquotient of lattices in a representation havingP.

ThenV hasP.

It is not hard to see that we have the implications (4)=⇒(3)=⇒(2)=⇒(1). Conjecture 1.0.1 is just (4) for P the property of being semi-stable or crystalline with bounded Hodge–Tate weights. For the same conditionP, (3) is established in [16], which is sufficient for applications to modularity theorems as in [15] (whereas (1) is not). Recently, Berger and Colmez proved (2) for Pthe property of being de Rham, crystalline or semi-stable with bounded Hodge–Tate weights via the theory of(ϕ,Γ)-modules in [1].

Convention 1.0.3. – We will deal with many p-power torsion modules. To simplify our notations, ifM is aZ-module, then we denoteM/pⁿM byMn. We also have to consider various Frobenius structures on different modules. To minimize possible confusion, we sometimes add a subscript toϕto indicate over which module the Frobenius is defined. For example,ϕMindicates the Frobenius defined overM. We often drop the subscript if no confusion will arise. We use contravariantfunctors (almost) everywhere. So removing the “ * ” from the notations for those functors will be more convenient. For example, the notationVstas used in this paper is denoted byV_st^∗in [7]. IfV is a finiteZp-representation ofG_∞, we denote byV^∨the dual representation ofV, i.e.,V^∨= Hom_Zp(V,Qp/Zp)ifV is killed by some power ofpandV^∨= Hom_Zp(V,Zp) ifV is a finite free Zp-module. Finally, ifX is a matrix,X^tdenotes its transpose. We always denote the identity map byId.

2. ϕ-modules of finiteE(u)-height and representations ofG_∞

This paper consists of 2 parts. §2–§4 is the first part, where we mainly discuss the theory of ϕ-modules of finiteE(u)-height overSand their associatedZp-representations ofG_∞. The key results to be proved are Theorem 2.4.2, Theorem 3.2.2 and Theorem 2.4.1 and its refinement Corollary 4.4.1. The second part (§5–§8) of this paper will combine the inputs from the first part and Kisin’s result (Theorem 5.4.1) to prove Conjecture 1.0.1.

2.1. Preliminaries

Throughout this paper we fix a positive integer rand a uniformiserπ∈K with Eisenstein polynomialE(u). Recall thatS=W(k)JuKis equipped with a Frobenius endomorphismϕvia u →u^pand the natural Frobenius onW(k). Aϕ-module(overS) is anS-moduleMequipped with a ϕ-semi-linear mapϕ:M→M. A morphism between two objects(M1, ϕ1),(M2, ϕ2)

is aS-linear morphism compatible with theϕi. Denote byMod^r_/Sthe category ofϕ-modules of finiteE(u)-height rin the sense thatMis of finite type overS and the cokernel ofϕ^∗ is killed byE(u)^r, whereϕ^∗ is theS-linear map1⊗ϕ:S⊗ϕ,SM→M. LetMod^r,tor_/S be the sub-category ofMod^r_/Sconsisting of finiteS-modulesMwhich are killed by some power of pand have projective dimension 1 in the sense thatMhas a two term resolution by finite free S-modules. We giveMod^r_/Sthe structure of an exact category induced by that on the Abelian category ofS-modules. We denote byMod^r,fr_/S the subcategory ofMod^r_/Sconsisting of finite freeS-modules. LetR= lim_←−OK^¯/pwhere the transition maps are given by Frobenius. By the universal property of the Witt vectorsW(R)ofR, there is a unique surjective projection map θ:W(R)→OK^¯ to thep-adic completion ofOK^¯, which lifts the projection R→ OK^¯/ponto the first factor in the inverse limit. Let πn∈K¯ be a pⁿ-root of π, such that (πn+1)^p=πn; write π= (πn)n0∈R and let [π]∈W(R) be the Teichmüller representative. We embed the W(k)-algebra W(k)[u] into W(R) by the map u →[π]. This embedding extends to an embedding S→W(R), and, asθ([π]) =π, θ|S is the mapS→ OK sendinguto π. This embedding is compatible with Frobenius endomorphisms. Denote byO_E thep-adic completion ofS[_u¹]. ThenOE is a discrete valuation ring with residue field the Laurent series ringk((u)).

We writeE for the field of fractions ofOE. IfFrRdenotes the field of fractions ofR, then the inclusionS→W(R)extends to an inclusionOE →W(FrR). LetE^ur⊂W(FrR)[¹_p]denote the maximal unramified extension ofE contained inW(FrR)[¹_p], andO^ur its ring of integers.

SinceFrRis easily seen to be algebraically closed, the residue fieldO^ur/pO^uris the separable closure ofk((u)). We denote byE^urthep-adic completion ofE^ur, and byO^urits ring of integers.

E^uris also equal to the closure ofE^urinW(FrR)[_p¹]. We writeS^ur=O^ur∩W(R)⊂W(FrR).

We regard all these rings as subrings of W(FrR)[_p¹]. Recall that K_∞=

n0K(π_n), and G_∞= Gal( ¯K/K_∞).G_∞acts continuously onS^urandE^urand fixes the subringS⊂W(R).

Finally, we denote byRep_Zp(G_∞)the category of continuousZp-linear representations ofG_∞ on finiteZp-modules and byRep^tor_Zp(G_∞)the subcategory consisting of those representations killed by some power ofp.

2.2. Fontaine’s theory on finiteZp-representations ofG_∞

Recall ([8], A, §1.1.4) that a finiteOE-module M is calledétale if M is equipped with a ϕ-semi-linear mapϕM:M→M, such that the inducedOE-linear mapϕ^∗_M:OE⊗ϕ,OEM →M is an isomorphism. We denote by ΦM_OE the category of étale modules with the obvious morphisms. An argument in [4], §2.1.1, shows thatK_∞/K is a strictly APF extension in the sense of [24]. Then Proposition A 1.2.6 in [8] implies that the functor

T^∨:ΦM_OE→Rep_Zp(G_∞); M →(M⊗_OEO^ur)^ϕ=1 (2.2.1)

is an equivalence of Abelian categories and the inverse ofT^∨is given by Rep_Zp(G_∞)→ΦM_OE; V →(V ⊗_ZpO^ur)^G∞.

In particular, for any M ∈ΦM_OE, we have the following natural O^ur-linear isomorphism compatible withϕ-structures.

˜

ι:M⊗OEO^urT^∨(M)⊗ZpO^ur. (2.2.2)

e ◦

We frequently use the contravariant version ofT^∨in this paper. ForM∈ΦM_OE, define T(M) = Hom_OE,ϕ(M,O^ur) ifMisp-torsion free

(2.2.3)

and (recallO^ur_n =O^ur/pⁿO^ur)

T(M) = Hom_OE,ϕ(M,O_n^ur) ifM is killed bypⁿ. (2.2.4)

It is easy to show thatT^∨(M)is the dual representation ofT(M). See for example §1.2.7 in [8], where Fontaine usesV_E^∗(M)to denoteT(M).

Recall that a S-moduleMis calledp-torsion free([8], B 1.2.5) if for all nonzero x∈M, Ann(x) = 0orAnn(x) =pⁿSfor somen. This is equivalent to the natural mapM→M⊗SOE

being injective. IfMis killed by some power ofp, thenMisp-torsion free if and only ifMis u-torsion free. Aϕ-moduleMoverSis calledétaleifMisp-torsion free andM⊗SO_E is an étaleOE-module. SinceE(u)is a unit inOE, we see that for anyM∈Mod^r_/S,Mis étale if and only ifMisp-torsion free. Obviously, any object inMod^r,fr_/Sis étale. In the next subsection, we will show that any object inMod^r,tor_/S is also étale. For any étaleM∈Mod^r,tor_/S , we can associate aZp[G_∞]-module via

TS(M) = HomS,ϕ

M,S^ur[1/p]/S^ur . (2.2.5)

Similarly, for anyM∈Mod^r,fr_/S, we define

TS(M) = HomS,ϕ(M,S^ur).

(2.2.6)

There is a natural injectionTS(M)→T(M)whereM:=M⊗SOE. In fact, this injection is an isomorphism by the following Proposition 2.2.1 below. LetΛbe aϕ-module overS. We denote byF_S(Λ)the set ofS-submodulesMsuch thatMis ofS-finite type, stable underϕand étale.

Definej_∗(Λ) =

M∈FS(Λ)M. IfAis a ring of characteristicp, we denote byA^septhe separable closure ofA.

PROPOSITION 2.2.1 (Fontaine). – For alln1, we have (1) j_∗(FrR) =k((π))^sep∩R=kJπK^sep,

(2) j_∗(Wn(FrR)) =S^ur_n,

(3) j_∗(W(FrR))⊂S^urandj_∗(W(FrR))is dense inS^ur, (4) S^ur_n =W_n(R)∩ O^urn ⊂W_n(FrR).

Proof. –Proposition 1.8.3 in [8]. Note that Fontaine usesA⁺_S,nto denoteS^ur_n. 2

COROLLARY 2.2.2. – Let M ∈ Mod^r,tor_/S be étale or M ∈Mod^r,fr_/S. Then TS(M) = T(M⊗SO_E).

Proof. –LetM:=M⊗SOE. It suffices to show that the natural injectionT_S(M)→T(M) is a surjection. Suppose that M is killed by pⁿ. For any f ∈T(M) = Hom_S,ϕ(M,O^ur_E,n), f(M)⊂ O_E^ur_,nis obviously aS-module ofS-finite type, stable underϕ. SinceO^ur_E,nis obviously p-torsion free, f(M) is p-torsion free. By Lemma 2.3.1 below, we see thatf(M) is étale.

Therefore f(M)∈FS(O^ur_E,n)andf(M)⊂S^ur. Thusf ∈HomS,ϕ(M,S^ur_n) =TS(M). The above proof also works ifMisS-finite free by replacingS^ur_n withS^ur, andO_E^ur_,nwithO^ur_E . 2

COROLLARY 2.2.3. – For alln1,S^ur_n[_u¹] =S^ur_n ⊗SOE On^ur.

Proof. –It is clear ifn= 1by Proposition 2.2.1 (1). The more general case can be proved by a standarddévissageargument; the details are left to the readers. 2

2.3. Some properties ofMod^r,tor_/S

By §2.3 in [14] and [4], if p3, thenMod^1,tor_/S is anti-equivalent to the category of finite flat group schemes overOK (also see [17]). It is thus expected that modules inMod^r,tor_/S have similar properties to those of finite flat group schemes overOK. In this subsection, we extend some basic properties of finite flat group schemes overOK toMod^r,tor_/S .

LEMMA 2.3.1. – Let0→M→M→M→0be an exact sequence ofϕ-modules overS.

Suppose thatM,MandMarep-torsion free andM∈Mod^r_/S. ThenMandMare étale and inMod^r_/S.

Proof. –See Proposition 1.3.5 in [8]. 2

PROPOSITION 2.3.2. – Let M ∈Mod^r_/S be killed by pⁿ. The following statements are equivalent:

(1) M∈Mod^r,tor_/S , (2) Misu-torsion free, (3) Mis étale,

(4) Mis a successive extension of finite freekJuK-modulesM_iwithM_i∈Mod^r_/S, (5) Mis a quotient of two finite freeS-modulesNandNwithN,N∈Mod^r,fr_/S.

Proof. –(1)=⇒(2) By the definition ofMod^r,tor_/S , there exist finite freeS-modulesN and Nsuch that

0→N→N→M→0

is exact. Letα₁, . . . , α_d andβ₁, . . . , β_d be bases forN andN respectively and let Abe the transition matrix; that is,(α₁, . . . , α_d) = (β₁, . . . , β_d)A. SinceMis killed by pⁿ for some n, there exists a matrixB with coefficients inSsuch thatAB=pⁿI. Now suppose thatx¯∈Mis killed byu^mwithx=d

i=1xiαi. Then we have

u^m(x₁, . . . , x_d) = (y₁, . . . , y_d)A^t for someyi∈S,i= 1, . . . , d. SinceAB=pⁿI, we have

(y1, . . . , yd) =u^m(pⁿ)⁻¹(x1, . . . , xd)B^t.

Let(z₁, . . . , z_d) = (pⁿ)⁻¹(x₁, . . . , x_d)B^t. Sincey_i∈S, it is not hard to see thatz_i∈Sfor all i= 1, . . . , d. Then we see that(x₁, . . . , x_d) = (z₁, . . . , z_d)A^t,x∈Nandx¯= 0.

(2)⇐⇒(3) SinceE(u)is a unit inOE, anyM∈Mod^r_/Sis étale if and only ifMisp-torsion free. IfMis killed by some power ofp, then this is equivalent toMbeingu-torsion free.

(3)=⇒(4) We proceed by induction onn. The casen= 1is obvious. Forn >1, consider the exact sequence of étaleOE-modules

0−→pM−→M−→^pr M/pM−→0,

whereM:=M⊗SOE andpris the natural projection. LetM= pr(M)andM= Ker(pr), then we get an exact sequence ofϕ-modules overS

0−→M−→M−→^pr M−→0.

(2.3.1)

e ◦

By induction, it suffices to show that M andM are étale and belong toMod^r_/S. But M andMare obviouslyu-torsion free and hencep-torsion free. Then by Lemma 2.3.1, we have M,M∈Mod^r_/S.

(4) =⇒ (5) Fontaine has proved this result (Theorem 1.6.1 in B, [8]) for the case e= 1.

In particular, Fontaine’s argument for reducing the problem to the case that M is killed by p also works here. Therefore, without loss of generality, we may assume that M is killed by p. In this case, M is a finite free kJuK-module. Let α1, . . . , αd be a basis of M and ϕ(α₁, . . . , α_d) = (α₁, . . . , α_d)X, where X is a d×d matrix with coefficients in kJuK. Since the cokernel of ϕ^∗_M is killed by u^er, there exists a matrix Y with coefficients in kJuK such thatXY =u^erI, whereIis the identity matrix. LetNbe a finite freekJuK-module with basis β₁, . . . , β_d, β₁, . . . , β_d and aϕ-structure defined by

ϕN(β1, . . . , βd, β₁, . . . , β_d) = (β1, . . . , βd, β₁, . . . , β_d) I 0 0 u^erI

A I I uI

,

whereA= (I−uY)⁻¹(ϕ(E)−Y)andE=X−u^er+1I. It is obvious that(N, ϕN)belongs to Mod^r,tor_/S . We construct aS-linear mapf:N→Mdefined by:

f(β1, . . . , βd, β₁, . . . , β_d) = (α1, . . . , αd)(E, I).

(2.3.2)

It is obvious that f is surjective. To check that f is compatible with ϕ-structures, it suffices to check f ◦ϕN=ϕM◦f on the basis. This is equivalent to verifying the following matrix equation:

X

ϕ(E), I

= (E, I) I 0 0 u^erI

A I I uI

,

which is a straightforward computation. So let N be a finite free S-module with basis βˆ1, . . . ,βˆd,βˆ₁, . . . ,βˆ_d and aϕ-structure defined by

ϕ_N( ˆβ₁, . . . ,βˆ_d,βˆ₁, . . . ,βˆ_d) = ( ˆβ₁, . . . ,βˆ_d,βˆ₁, . . . ,βˆ_d) I 0 0 E(u)^rI

Aˆ I I uI

with Aˆ any lift of A. It is easy to check that N=N/pN andN∈Mod^r,fr_/S. Thus we have a ϕ-module morphism g:N→M with g surjective. Let N= Ker(g). Using the explicit definition (2.3.2) off, we can easily find aS-basis forN. ThusNisS-finite free. Finally, using Lemma 2.3.1 for0→N→N−→^g M→0, we see thatN∈Mod^r,fr_/S.

(5)=⇒(1) Trivial. 2

COROLLARY 2.3.3. – Letf:M→M be a morphism inMod^r,tor_/S . ThenKer(f)belongs to Mod^r,tor_/S .

Proof. –Lemma 2.3.1 shows that Ker(f)∈Mod^r_/S and Ker(f) is obviously u-torsion free. 2

In general,Cok(f)is not necessarily inMod^r,tor_/S . See Example 2.3.5.

By the above lemma, any object M∈Mod^r,tor_/S is étale. Thus Corollary 2.2.2 implies that T_S(M) = T(M⊗SOE) = Hom_OE,ϕ(M⊗SOE,O^ur). Therefore, the functor T_S defined in (2.2.5) is well defined onMod^r,tor_/S . In summary, we have

COROLLARY 2.3.4. – The contravariant functorTS fromMod^r,tor_/S toRep^tor_Zp(G_∞)is well defined and exact.

Ifr= 1andp >2, [4] and §2.3 in [14] proved that there exists an anti-equivalenceGbetween Mod^1,tor_S (resp.Mod^1,fr_S ) and the category of finite flat group schemes overOK(resp.p-divisible groups over OK). Furthermore, for any M∈Mod^1,tor_S (resp.Mod^1,fr_S ), there exists a natural isomorphism ofZp[G_∞]-modules

G(M)( ¯K)|G∞T_S(M).

In general,T_Sis not fully faithful iferp−1.

Example 2.3.5. – Let S:=S·α be the rank-1 free S-module equipped with ϕ(α) = c0−1E(u)·αwherepc0is the constant coefficient ofE(u). By Example 2.2.3 in [4], ifp >2, G(S) =μp^∞. In particular,TS(S) =μp^∞( ¯K)|G_∞=Zp(1). Ifp= 2, Theorem (2.2.7) in [14]

shows thatG(S)is isogenous toμ₂∞. ThusT_S(S)is a G_∞-stableZ2-lattice inQ2(1). So we still haveTS(S)Z2(1). Suppose e=p−1. Consider the map f:S₁→S₁ given by α →c⁻¹₀ u^e. An easy calculation shows that fis a well-defined morphism of ϕ-modules and f⊗SO_E is an isomorphism. ThenTS(f)is an isomorphism butfis not. Also,Cok(f)is not an object inMod^r,tor_/S .

The following lemma is an analogy of “scheme-theoretic closure” in the theory of finite flat group schemes overOK.

LEMMA 2.3.6 (Scheme-theoretic closure). – Let f:M→L be a morphism of ϕ-modules over S. Suppose that Mand Lare p-torsion free andM∈Mod^r_/S. PutM= Ker(f)and M=f(M). ThenMandMare étale and belong toMod^r_/S. In particular, ifM∈Mod^r,tor_/S , thenMandM∈Mod^r,tor_/S .

Proof. –By the construction, it is obvious thatMandMarep-torsion free. By Lemma 2.3.1, M andMare étale and belong toMod^r_/S. IfM∈Mod^r,tor_/S , thenM andM areu-torsion free. By Proposition 2.3.2 (2), we see thatMandMbelong toMod^r,tor_/S . 2

LEMMA 2.3.7. – LetM∈Mod^r_/Sbe torsion free,M=M⊗SOE. Then there exists a finite freeS-moduleM∈Mod^r,fr_/S such thatM⊂M⊂M.

Proof. –LetM=M ∩M[1/p]. By Proposition B 1.2.4 of [8], we haveM⊂M⊂M with Ma finite freeS-module. It is obvious thatMisϕ-stable, so it remains to check thatCok(ϕ^∗_M) is killed by E(u)^r. Note that there exists an integer s such that p^sM ⊂M. Since E(u)^r killsCok(ϕ^∗_M), we have that p^sE(u)^r killsCok(ϕ^∗_M). Letα1, . . . , αd be a basis ofM and ϕM(α1, . . . , αd) = (α1, . . . , αd)AwhereAis ad×dmatrix with coefficients inS. SinceM is étale,A⁻¹exists with coefficients inOE. It suffices to prove thatE(u)^rA⁻¹has coefficients inS, but this follows easily from the fact thatp^sE(u)^rA⁻¹has coefficients inS. 2

COROLLARY 2.3.8. – Letf:M→Nbe a surjective morphism inMod^r_/SwithM∈Mod^r,fr_/S a finite freeS-module andN∈Mod^r,tor_/S killed by some power ofp. ThenL:= Ker(f)∈Mod^r,fr_/S isS-finite free.

Proof. –By Lemma 2.3.1, L∈Mod^r_/S. L is obviously torsion free and of S-finite type.

Let L:=L⊗SOE and M :=M⊗SOE. Since N is u-torsion free, we have M∩L=L.

By the proof of Lemma 2.3.7, we see that L∩L[¹_p] is S-finite free. But L[¹_p] =M[¹_p], so L∩L[¹_p] =L∩(M ∩M[¹_p]) =L∩M=L. ThusLisS-finite free. 2

e ◦

COROLLARY 2.3.9. – LetM∈Mod^r,fr_/S (resp.Mod^r,tor_/S ), letNbe aϕ-stableS-submodule ofMandN:=N⊗SOE. Then there exists anN∈Mod^r,fr_/S (resp.N∈Mod^r,tor_/S )such that N⊂N⊂N∩M.

Proof. –Let M =M⊗SOE and L=M/N. By Lemma 2.3.6, in this case, there exists N∈Mod^r_/S such that N⊂N⊂M∩N and N is étale. If M is inMod^r,tor_/S , then put N=N, soN belongs to Mod^r,tor_/S becauseN is obviouslyu-torsion free. IfMis a finite free S-module, then N∈Mod^r_/S is torsion free. Therefore, by Lemma 2.3.7, there exists N∈Mod^r,fr_/S such thatN⊂N⊂N⊂M∩N. 2

2.4. Main results of the first part

Now we can state the main theorems to be proved in the first part of this paper. The first theorem is an analog of Raynaud’s theorem (Proposition 2.3.1 in [22]) which states that a Barsotti–Tate groupH overKcan be extended to a Barsotti–Tate group overOKif and only if, for eachn,H[pⁿ]can be extended to a finite flat group schemeHnoverOK.

THEOREM 2.4.1. – LetTbe a finite freeZp-representation ofG_∞. If for eachn, there exists an M_(n)∈Mod^r,tor_/S such that TS(M_(n))T /pⁿT, then there exists a finite free S-module M∈Mod^r,fr_/S such thatTS(M)T.

Though the functorTSonMod^r,tor_/S is not a fully faithful functor iferp−1as explained in Example 2.3.5, we will prove that the functorTSenjoys “weak” full faithfulness.

THEOREM 2.4.2. – LetM,M∈Mod^r,tor_/S , letf:TS(M)→TS(M)be a morphism of finite Zp[G_∞]-modules. Then there exists a morphismf:M→M such thatT_S(f) =p^cf, wherecis a constant depending only on the absolute ramification indexe=e(K/K₀)and the heightr. In particular,c= 0ifer < p−1.

Remark 2.4.3. – The constantchas an explicit (but complicated) formula. We do not optimize it, so there should still be room to improve. We have proved a similar, though weaker, result in [19] for truncated Barsotti–Tate groups (see also [3]). The constant obtained here is independent of the height of the truncated Barsotti–Tate group, though we do use many of the techniques found in [19].

To prove these theorems, we need to construct the Cartier dual on Mod^r,tor_/S and a theorem (Theorem 3.2.2) to compareMwithTS(M). These preparations will be discussed in §3.

2.5. Construction ofS^f(r)

For a fixed heightr,S^uris too big to work with. In this subsection, we cut out aS-submodule S^f(r) inside S^ur which is big enough for representations arising from Mod^r_/S. Let Λ be a p-torsion freeϕ-module overS. We denote byF_S^fr(Λ)the set ofS-submodulesMofΛsuch thatM∈Mod^r_/S. SinceΛisp-torsion free,Mis étale, soF_S^fr(Λ)⊂FS(Λ). (Recall thatFS(Λ) is the set ofS-submodulesMsuch thatMis ofS-finite type and stable underϕ.) Define

S^f(r)_(n)=

M∈FS^fr(S^ur_n)

M for each fixedn1,

and

S^f(r)=

M∈F_S^fr(S^ur)

M.

Obviously,S^f(r)_(n) (resp.S^f(r)) is a subset ofS^ur_n (resp.S^ur).

PROPOSITION 2.5.1. – For eachn1,

(1) S^f(r)_(n) is aG_∞-stable andϕ-stableS-submodule ofS^ur_n, (2) S^f(r)is aG_∞-stable andϕ-stableS-submodule ofS^ur, (3) S^f(r)_(n) =S^f(r)/pⁿS^f(r), i.e.,S^f(r)_n =S^f(r)_(n).

Proof. –For each fixedn1, letMandM∈F_S^fr(S^ur_n). To prove (1), it suffices to check that M:=M+M∈F_S^fr(S^ur_n). It is obvious that S^f(r)_(n) is G_∞-stable and ϕ-stable. Since S^ur_n is p-torsion free, M is p-torsion free. It therefore suffices to check that the cokernel of ϕ^∗_M:S⊗ϕ,SM→M is killed by E(u)^r. This follows from the fact that the cokernels of ϕ^∗_M and ϕ^∗_M are killed byE(u)^r. The above argument also works for proving (2). For (3), we need to show that the natural map ι:S^f(r)_n →S^f(r)_(n) induced bypr :S^ur→S^ur_n is an isomorphism. We first prove the surjectivity by claiming that for any M∈F_S^fr(S^ur_n) there exists an N∈F_S^fr(S^ur)such that pr(N) =M. In fact, by Proposition 2.3.2, (3), there exists a finite free ϕ-module N ∈Mod^r,fr_/S with a surjection f:N M. Recall that the functor TS: Mod^r,tor_/S →Rep_Zp(G_∞) is exact (Corollary 2.3.4). Thus TS(f) :TS(N)→TS(M) is surjective, so by Lemma 2.2.2, there exists a morphism of ϕ-modules h:N→S^ur which lifts the identity embedding M→S^ur_n. Therefore N=h(N)∈F_S^fr(S^ur)and pr(N) =M, as required. For the injectivity, it suffices to prove that for anyM∈F_S^fr(S^ur)andx∈S^ur, if px∈M, then there exists L∈F_S^fr(S^ur)such thatx∈L. LetN be theS-submodule inS^ur generated by{ϕ^m(x)}m0andN˜ theS-submodule ofMgenerated byϕ^m(px). Letα: ˜N→N be the morphism defined by

α:

siϕ^mi(px) →

siϕ^mi(x).

SinceS^uris torsion free,αis an isomorphism andαextends to an isomorphismN˜ ⊗SO_E−→^∼ N⊗SO_E insideO^ur. By Corollary 2.3.9, we haveN˜∈Mod^r,fr_/Ssuch thatN˜ ⊂N˜⊂N˜⊗SO_E. LetL=α( ˜N). We see thatx∈N⊂L⊂O^ur withL∈Mod^r,fr_/S, so by Proposition 2.2.1 (3), x∈N⊂L⊂S^ur. 2

COROLLARY 2.5.2. – For eachn1,S^f(r)_n is flat overS_n.

Proof. –By Proposition 2.5.1 (3), it suffices to prove that, for anyM∈F_S^fr(S^ur), there exists a finite freeS-modulesM∈F_S^fr(S^ur)such thatM⊂M. By Lemma 2.3.7, there exists such a moduleM⊂M⊗SOE ⊂O^ur. By Proposition 2.2.1 (3), we see thatM⊂S^ur. 2

COROLLARY 2.5.3. – For anyM∈Mod^r,tor_/S , Hom_S,ϕ

M,S^f(r)⊗Z(Qp/Zp)

Hom_S,ϕ(M,E/O_E^ur) =T_S(M).

e ◦

3. A theorem to compareMwithTS(M)

In this section, we prove a “comparison” theorem (Theorem 3.2.2) to compare M with TS(M). This theorem will be the technical hearts in many of our proofs. In the following two sections, we will focus on torsion objectsMod^r,tor_/S . ForM∈Mod^r,tor_/S ,nwill always denote an integer such thatpⁿkillsM.

3.1. Cartier dual

We need to generalize toMod^r,tor_/S the concept ofCartier dualon finite flat group schemes overOK. Example 2.3.5 shows that ifr= 1, thenSis the correct Cartier dual ofS. Motivated by this example, we have:

Convention 3.1.1. – Define a ϕ-semi-linear morphism ϕ^∨:S→S by1 →c^−r₀ E(u)^r. We denote by S^∨ the ringSwithϕ-semi-linear morphismϕ^∨. The same notations apply forS_n andS^f(r)n , etc. By Example 2.3.5, we haveT_S(S^∨_n)Zp/pⁿZp(r).

Obviously, such “Cartier dual” (if it exists) must be compatible with the associated Galois representations, so we first analyze the dual on ΦM^tor_OE. Let M ∈ ΦM^tor_OE and M^∨ = Hom_OE(M,E/OE). As anOE-module, we haveMd

i=1OE,ni, so there exists a canonical perfect pairing ofOE-modules

,:M×M^∨→ E/OE. (3.1.1)

We equipE/OE with aϕ-structure by1 →c⁻₀^rE(u)^r. We will construct aϕ-structure onM^∨ such that (3.1.1) is also compatible withϕ-structures. AS-linear mapf:M→Nis also called ϕ-equivariantiff is a morphism ofϕ-modules.

LEMMA 3.1.2. – There exists a uniqueϕ-semi-linear morphismϕM^∨:M^∨→M^∨such that (1) (M^∨, ϕ^∨)∈ΦM^tor_OE.

(2) For anyx∈M, y∈M^∨,ϕM(x), ϕM^∨(y)=ϕ(x, y).

(3) T(M^∨)T^∨(M)(r)asZp[G_∞]-modules.

Proof. –We first construct aϕM^∨ satisfying (2). LetM d

i=1OE,niαi and letβ1, . . . , βd

be the dual basis of M^∨. WriteϕM(α1, . . . , αd) = (α1, . . . , αd)A, whereAis ad×dmatrix with coefficients inOE. Define

ϕM^∨(β1, . . . , βd) = (β1, . . . , βd)

c⁻₀^rE(u)^r (A⁻¹)^t.

Note thatAis invertible inO_E becauseM is étale. It is easy to check that(M^∨, ϕM^∨)satisfies (1), (2) and uniqueness, so it remains to check (3). We can extend the ϕ-equivariant perfect pairing,to

,: (M⊗_OEO^ur)×(M^∨⊗_OEO^ur)→ O^ur,∨_n , (3.1.2)

wheren= Max(n1, . . . , nd). Since the above pairing isϕ-equivariant, we have a pairing (M⊗_OEO^ur)^ϕ=1×(M^∨⊗_OEO^ur)^ϕ=1→(O^ur,_n ^∨)^ϕ=1.

Thus, we have a pairing

T^∨(M)×T^∨(M^∨)→Z/pⁿZ(−r) (3.1.3)