On a relation of overconvergence and F -analyticity on p -adic Galois representations of a p -adic field F
aa r X i v : . [ m a t h . N T ] S e p On a relation of overconvergence and F -analyticity on p -adic Galois representations ofa p -adic field F Megumi Takata ∗ September 17, 2019
Abstract
Let p be a prime number. There are properties called “overconver-gence” and “ F -analyticity” for p -adic Galois representations of a p -adicfield F . By Berger’s work, it is known that F -analyticity is stricter thanoverconvergence. In this article, we show that, in many cases, an overcon-vergent Galois representation is F -analytic up to a twist by a character.This result emphasizes the necessity of the theory of ( ϕ, Γ)-modules overthe multivariable Robba ring, by which we expect to study all p -adic Ga-lois representations. In the p -adic local Langlands program for GL ( Q p ), which has finally beenestablished in [Col10] and [CDP14], an important result in the early stage isa theorem due to Cherbonnier and Colmez [CC98]. It says that all of the p -adic Galois representations of a p -adic field are overconvergent with respectto the cyclotomic Z p -extension. It enables us to study every p -adic Galoisrepresentation of a p -adic field in the terms of modules over the univariableRobba ring, which is the ring of (not necessarily bounded) functions on p -adicannuli. One of the benefits appears in Berger’s work [Ber02], which relates thefollowing three notions: 1) ( ϕ, Γ)-modules, 2) p -adic differential equations, and3) filtered ( ϕ, N )-modules.Thus, if we want to generalize the p -adic local Langlands program fromGL ( Q p ) to GL ( F ) for an arbitrary p -adic field F , a similar question occurs:is any p -adic Galois representation of F overconvergent if we consider a generalLubin-Tate extension instead of the cyclotomic one? For this, Fourquaux and ∗ Kyushu Sangyo University, 3-1 Matsukadai 2-chome, Higashi-ku, Fukuoka 813-8503Japan.E-mail address: [email protected]
Mathematics Subject Classification. p -adic Galois representations which is notoverconvergent with respect to a Lubin-Tate extension. Then we face a nextquestion: how strict is overconvergence in a general Lubin-Tate setting? Berger[Ber16] has proved that, an another property called F -analyticity is a sufficientcondition for overconvergence. In addition, he has proved the following: Theorem 1.1 ([Ber13, Cor. 4.3]) . Any absolutely irreducible and overconver-gent p -adic Galois representations of F are F -analytic up to twist by a character. We remark that Berger and Fourquaux [BF17, Th. 1.3.1, Cor. 1.3.2] describean arbitrary overconvergent representation by using an F -analytic one and onewhich factors through Γ, where Γ is the Galois group of the considering Lubin-Tate extension.The main theorem of this article is a generalizaion of Theorem 1.1. To stateit in detail, we need to introduce some notations. We fix an algebraic closure F of F and write G F = Gal( F /F ). Let L ⊂ F denote a finite extension of F . Let F LT ̟ be the Lubin-Tate extension of F with respect to a fixed uniformizer ̟ ∈ F .We write Γ = Gal( F LT ̟ /F ). Let q be the order of the residue field of F . Let R be the Robba ring with coefficients in L . It has an action of Γ and ϕ q , where ϕ q is a lift of the q -th power map. We have a fully faithful functor R ⊗ E † D † from the category of L -representations of G F to that of the ( ϕ q , Γ)-modulesover R (for more precise descriptions of R , the notion of ( ϕ q , Γ)-modules, and
R ⊗ E † D † , see § Theorem 1.2.
Let V be an overconvergent L -representation of G F . Supposethat D = R ⊗ E † D † ( V ) has a filtration D ⊂ D ⊂ · · · ⊂ D r = D of ( ϕ q , Γ) -modules over R . We put ∆ i = D i /D i − . We assume that the follow-ing conditions hold. (a) . For any ≤ i ≤ r, we have End ( ϕ q , Γ)-mod / R (∆ i ) = L . (b) . For any ≤ i < j ≤ r , we have Hom ( ϕ q , Γ)-mod / R (∆ j , ∆ i ) = 0 . (c) . For any < i ≤ r, the short exact sequence → D i − → D i → ∆ i → does not split.Then there exist a finite extension L ′ of L and a character δ : G × F → ( L ′ ) × suchthat V ⊗ L L ′ ( δ ) is F -analytic. Remark 1.3. (i). The case r = 1 of Theorem 1.2 says that, if an overcon-vergent L -representation V of G F satisfies End( V ) = L , then V becomes F -analytic after twisting by a character. It is still a generalization ofTheorem 1.1. 2ii). Off course, there are overconvergent representations which can not be F -analytic after twisting by any character: for example, the direct sum of F -analytic one and not F -analytic one. However, a large part of overcon-vergent representations seem to satisfy the conditions of Theorem 1.2: forexample, most of trianguline representations. While, F -analyticity is verystrict since, by definition, it demands the Hodge-Tate weights with respectto any τ ∈ Hom(
F, F ) r { id F } to be zero. Hence this result predicts thatthe size of the class of overconvergent representations is thought to bevery small in the whole. Therefore we seem to need the theory of ( ϕ, Γ) -modules over the multivariable Robba ring ([Ber13], [Ber16], [BF17]), bywhich we expect to study all p -adic Galois representations.In §
2, we recall some definitions and theorems on the Lubin-Tate ( ϕ q , Γ)-modules. In particular, we define overconvergence and F -analyticity here. In §
3, after introducing several lemmas, we prove Theorem 3.4. Then we obtainTheorem 1.2 as a corollary.
Acknowledgments
The author would like to thank Kentaro Nakamura for many useful comments,in particular, for suggesting him to generalize the original version of the maintheorem. ( ϕ q , Γ) -modules In this section, we recall some notion on the Lubin-Tate ( ϕ q , Γ)-modules.Let F be a finite extension of Q p , O F the ring of integers of F , and q theorder of the residue field of O F . We fix an algebraic closure F of F . We write G F = Gal( F /F ).We fix a uniformizer ̟ of F and a power series f ̟ ( T ) ∈ O F [[ T ]] such that f ̟ ( T ) ≡ ̟T modulo degree 2 and f ̟ ( T ) ≡ T q modulo ̟ . Then there existsa unique formal group law G ̟ ( X, Y ) ∈ O F [[ X, Y ]] such that f ̟ ∈ End( G ̟ ),which we call the Lubin-Tate formal group associated to ̟ . It has a naturalformal O F -module structure [ · ] : O F → End( G ̟ ) such that, for any a ∈ O F ,the first term of [ a ]( T ) is aT and [ ̟ ]( T ) = f ̟ ( T ). For any a ∈ O F , we put G ̟ [ a ] = { α ∈ F | [ a ]( α ) = 0 } . We denote by F LT ̟ the extension of F in F obtained by adding all elements of G ̟ [ ̟ n ] for all integers n ≥
1. We call F LT ̟ the Lubin-Tate extension of F associated to ̟ . We define the Tate module T G ̟ of G ̟ by T G ̟ = lim ←− n G [ ̟ n ] . This is a free O F -module of rank 1 on which G F naturally acts. It induces a character χ LT : G F → O × F , which we call theLubin-Tate character associated to ̟ . We write H F = Gal( F /F LT ̟ ), which isthe kernel of χ LT . For any integer n ≥
1, we putΓ = Gal( F LT ̟ /F ) = G F / H F ∼ −→ O × F andΓ n = Gal( F LT ̟ /F ( G ̟ [ ̟ n ])) . L ⊂ F be a finite extension of F . We put O E = O E L = (X n ∈ Z a n T n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n ∈ O L for any n ∈ Z , a n → n → −∞ ) ) , which is a discrete valuation ring such that a uniformizer ̟ L of L generates themaximal ideal. We put E = E L = Frac( O E ). In this article, we consider theweak topology . For any n ∈ Z , we define a topology on ̟ nL O E with { ̟ i + nL O E + T j ̟ nL O L [[ T ]] } i,j as a fundamental system of neighborhoods of 0. We define atopology on E = ∪ n ∈ Z ̟ nL O E by the inductive limit topology.An L -linear ( ϕ q , Γ)-action on O E or on E is defined such that, for any γ ∈ Γ,we have ϕ q ( T ) = f ̟ ( T ) , γ.T = [ χ LT ( γ )]( T ) . A free O E -module D of finite rank is called a ( ϕ q , Γ) -module over O E if:1. Γ acts continuously and semilinearly on D and2. a ϕ q -semilinear map Φ : D → D is equipped such that • the action of Γ and Φ commute, and • the O E -linear homomorphism O E ⊗ ϕ q , O E D → D induced by Φ isan isomorphism.We abuse notation and use the same symbol ϕ q to denote Φ . The notion of( ϕ q , Γ) -modules over E is defined in a similar way as above. We call a ( ϕ q , Γ)-module D over E ´etale if there exists a ( ϕ q , Γ)-module D over O E such that D ≃ E ⊗ O E D as ( ϕ q , Γ)-modules over E .We denote by Rep L ( G F ) the category of continuous finite dimensional L -representations of G F and by Mod ϕ q , Γ / E the category of ( ϕ q , Γ)-modules over E . Let Mod ϕ q , Γ , ´et / E denote the full subcategory of Mod ϕ q , Γ / E consisting of ´etaleobjects. Fontaine have found an equivalence of categories of Rep L ( G F ) andMod ϕ q , Γ , ´et E in the cyclotomic case. In the general Lubin-Tate cases, it is givenby Kisin and Ren. To construct it, we use a big field B , which contains E F andhas a ( ϕ q , G F )-action. Moreover, we have B H F = E F . For a precise definitionof B , the reader may refer to [KR09] or [FX13, 1B]. Note that, in [KR09] (resp.[FX13]), this is denoted by b E ur (resp. B). Theorem 2.1 ([Fon91, Th. 3.4.3, Rem. 3.4.4], [KR09, Th. 1.6]) . We have afunctor D : Rep L ( G F ) → Mod ϕ q , Γ , ´et / E : V ( V ⊗ F B ) H F , which gives an equivalence of categories. Its quasi-inverse functor is given by D ( D ⊗ E F B ) ϕ q =1 . Now we define a subfield E † of E and a ring R containing E † . Let v p denotethe p -adic additive valuation on F normalized as v p ( p ) = 1. For any formal4eries f ( T ) = P n ∈ Z a n T n with coefficients in L and any real number r ≥
0, wedefine v { r } ( f ) = inf n v p ( a n ) + nr , which may be ±∞ . For any 0 ≤ s ≤ r , weput v [ s,r ] ( f ) = inf s ≤ r ′ ≤ r v { r ′ } ( f ). We define E [ s,r ] = E [ s,r ] L = ( f ( T ) = X n ∈ Z a n T n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n ∈ L for any n ∈ Z ,v [ s,r ] ( f ) = −∞ ) , E ]0 ,r ] = E ]0 ,r ] L = \ E ]0 ,r ] , E (0 ,r ] = E (0 ,r ] L = E ]0 ,r ] ∩ E , and E † = E † L = [ r> E (0 ,r ] . They have ring structure and Γ acts on them such that γ.T = [ χ LT ( γ )]( T ) forany γ ∈ Γ. Moreover, if r is sufficiently small, then we have a ring endomorphism ϕ q : E [ s,r ] → E [ sq,rq ] such that ϕ q ( T ) = f ̟ ( T ). Hence the rings E † and R areendowed with a ( ϕ q , Γ)-action. We call R the Robba ring and E † the boundedRobba ring . We have inclusions E ⊃ E † ⊂ R as rings with a ( ϕ q , Γ)-action.The ring E [ s,r ] is equipped with the topology defined by the valuation v [ s,r ] .Then E [ s,r ] is complete. We endow E ]0 ,r ] with the projective limit topology, and R with the inductive limit topology. On E (0 ,r ] , we consider the topology definedby the valuation v { r } . We endow E † with the inductive limit topology.The notion of ( ϕ q , Γ) -modules over E † or R is defined in a similar way asthose over O E or E . A ( ϕ q , Γ)-modules D † over E † is called ´etale if D † ⊗ E † E is´etale. A ( ϕ q , Γ)-modules D over R is called ´etale or of slope 0 if there existsan ´etale ( ϕ q , Γ)-module D † over E † such that D ≃ R ⊗ E † D † . Let Mod ϕ q , Γ / E † (resp. Mod ϕ q , Γ / R ) denote the category of ( ϕ q , Γ)-modules over E † (resp. R ). Wedenote by Mod ϕ q , Γ , ´et / E † (resp. Mod ϕ q , Γ , ´et / R ) the full subcategory of Mod ϕ q , Γ / E † (resp.Mod ϕ q , Γ / R ) consisting of ´etale objects. Theorem 2.2 ([FX13, Prop. 1.6]) . The functor D †
7→ R ⊗ E † D † gives anequivalence of categories of Mod ϕ q , Γ , ´et / E † and Mod ϕ q , Γ , ´et / R . We have a subfield B † of B as in [FX13, 1B], which contains E † F and hasa ( ϕ q , G F )-action. Moreover, we have ( B † ) H F = E † F . For any object V ofRep L ( G F ), we put D † ( V ) = ( V ⊗ F B † ) H F . We have a natural inclusion D † ( V ) ⊂ D ( V ). Definition 2.3.
An object V of Rep L ( G F ) is overconvergent if D † ( V ) gener-ates D ( V ) as an E -vector space. We denote by Rep oc L ( G F ) the subcategory ofRep L ( G F ) consisting of overconvergent objects.5 heorem 2.4 ([FX13, Prop. 1.5]) . The operation D † gives an equivalence ofcategories of Rep oc L ( G F ) and Mod ϕ q , Γ , ´et E † . Moreover, the diagram Rep oc L ( G F ) D † / / (cid:127) _ (cid:15) (cid:15) Mod ϕ q , Γ , ´et / E † E⊗ E† ( · ) (cid:15) (cid:15) Rep L ( G F ) D / / Mod ϕ q , Γ , ´et / E commutes up to canonical isomorphisms. Remark 2.5.
Cherbonnier and Colmez [CC98] have shown that, in the cyclo-tomic case, we have Rep oc L ( G F ) = Rep L ( G F ), i.e. all of the L -representationsof G F are overconvergent with respect to the cyclotomic extension. Even inthe general Lubin-Tate case, all of the 1-dimensional L -representations of G F are overconvergent [FX13, Remark 1.8]. However, there exist L -representationsof G F which are not overconvergent, as shown by Fourquaux and Xie [FX13,Theroem 0.6].Now, let us recall another property of L -representations of G F so-called F -analyticity. We write C p for the p -adic completion of F . Definition 2.6.
An object V of Rep L ( G F ) is called F -analytic if, for any τ ∈ Hom Q p -alg ( F, F ) r { id F } , the C p -representation C p ⊗ τ,F V is trivial.For F -analytic representations, Berger shows the following: Theorem 2.7 ([Ber16, Thm. C]) . Any F -analytic L -representation of G F isoverconvergent. There is also a notion of F -analyticity for ( ϕ q , Γ)-modules over R . To defineit, we introduce an action of a Lie algebra. Let Lie Γ denote the Lie algebraassociated to the p -adic Lie group Γ. Note that there is an isomorphism Lie Γ ∼ −→O F induced by χ LT . Let D be any object in Mod ϕ q , Γ / R . We will define an actionof Lie Γ on D . Take any x ∈ D . If β ∈ Lie Γ is sufficiently close to 0, then wecan define γ = exp β ∈ Γ. Now we recall the following:
Lemma 2.8 ([FX13, Lemma 1.7]) . For any r > and < s ≤ r , there exists n = n ( s, t ) such that, for any γ ′ ∈ Γ n and f ∈ E [ s,r ] , we have v [ s,r ] (( γ ′ − f ) ≥ v [ s,r ] ( f ) + 2 . Note that, in the original version, the right hand side is v [ s,r ] ( f )+1. However,by the proof, we can replace 1 with any other positive real number, for example2. We fix a basis e , . . . , e d of D . For any r >
0, we put D ]0 ,r ] = ⊕ di =1 E ]0 ,r ] e i . It depends on the choice of e , . . . , e d . However, for another choice e ′ , . . . , e ′ d ,there exists 0 < r ′ < r such that ⊕ di =1 E ]0 ,r ′ ] e i = ⊕ di =1 E ]0 ,r ′ ] e ′ i . We put D [ s,r ] = E [ s,r ] ⊗ E ]0 ,r ] D ]0 ,r ] . orollary 2.9. For any r > and < s ≤ r , there exists n = n ( s, t ) such that,for any γ ′ ∈ Γ n and x ∈ D [ s,r ] , we have v [ s,r ] (( γ ′ − x ) ≥ v [ s,r ] ( x ) + 2 .Proof. Since we can prove it in the same way as [Ber02, Lem. 5.2], we omitit. We choose an integer m ≥ γ p m ∈ Γ n ( s,t ) . By Corollary 2.9, theseries (log γ ) [ s,r ] ( x ) = 1 p m ∞ X i =1 ( − i − ( γ p m − i i x converges in D [ s,r ] , and it is independent of the choice of m . We can show that, if s ′ < s , then the image of (log γ ) [ s ′ ,r ] ( x ) via the natural map D [ s ′ ,r ] → D [ s,r ] coin-cides with (log γ ) [ s,r ] ( x ). Thus we obtain a projective system ((log γ ) [ s,r ] )
An object D of Mod ϕ q , Γ / R is called F -analytic if the operator ∇ β | D is independent of the choice of β ∈ Lie Γ. If so, we often denote ∇ β | D by ∇| D , or ∇ if no confusion occurs.We have a formula to calculate log γ as follows: Lemma 2.11.
For any x ∈ D and an element γ ∈ Γ sufficiently close to 1, wehave (log γ ) x = lim n →∞ γ p n x − xp n . Proof.
It suffices to show the formula for D [ s,r ] . We fix a basis e , . . . , e d of D . For any y = P di =1 y i e i ∈ D [ s,r ] = ⊕ di =1 E [ s,t ] e i , we define v [ s,r ] ( y ) =inf ≤ i ≤ d v [ s,r ] ( y i ). Let L be an L -linear operator on D [ s,r ] . We define a val-uation of L by v [ s,r ] ( L ) = inf y ∈ D [ s,r ] ( v [ s,r ] ( L y ) − v [ s,r ] ( y )) . If v [ s,r ] ( L ) > ( p − − , then the operatorexp L = ∞ X i =0 L i i !on D [ s,r ] is well-defined.By corollary 2.9, if an element γ ∈ Γ is sufficiently close to 1, then theoperator log γ on D [ s,r ] is well-defined and v [ s,r ] (log γ ) ≥
2. Thus we can defineexp(log γ ). Moreover, we have γx = [exp(log γ )]( x ) . n ≥
1, we have γ p n x = [exp( p n log γ )]( x )= ∞ X i =0 ( p n log γ ) i i ! x, and γ p n x − xp n = (log γ )( x ) + p n ∞ X i =0 p ni (log γ ) i +2 ( i + 2)! x. (1)Note that the second term of the right-hand side of (1) is well-defined since v [ s,r ] (cid:0) p ni (log γ ) i +2 x/ ( i + 2)! (cid:1) tends to 0 as i → ∞ . As n goes to ∞ in (1), wehave the required equality.Berger shows that F -analyticity of an L -representation of G F is equivalentto that of the corresponding ( ϕ q , Γ)-module.
Theorem 2.12 ([Ber16, Thm. D]) . Let V be any object of Rep oc L ( G F ) . Then V is F -analytic if and only if R ⊗ E † D † ( V ) is F -analytic. In this section, we introduce several lemmas first. Next, we prove Theorem3.4, which is a ( ϕ q , Γ)-module version of the main theorem. Then we obtainTheorem 1.2 as a corollary.Let D and D be ( ϕ q , Γ)-modules over R . We denote by Ext( D , D ) theset of isomorphism classes of extensions of D by D in the category of ( ϕ q , Γ)-modules over R , which has a natural L -vector space structure. Lemma 3.1.
Let ∆ and D be ( ϕ q , Γ) -modules over R . Suppose that D is F -analytic and End ( ϕ q , Γ)-mod / R (∆) = L. If Ext(∆ , D ) = 0 , then ∆ is F -analytic.Proof. This lemma is a generalization of [FX13, Theorem 5.20]. Suppose that ∆is not F -analytic. Then there exist β, β ′ ∈ Lie Γ such that ∇ β | ∆ = ∇ β ′ | ∆ . Weput ∇ ′ | ∆ = ∇ β | ∆ − ∇ β ′ | ∆ . Since ∇ β | ∆ and ∇ β ′ | ∆ are both R -derivations andthe trivial ( ϕ q , Γ)-module R is F -analytic, the operator ∇ ′ | ∆ is R -linear. More-over, it is stable under ( ϕ q , Γ)-action. Hence we have ∇ ′ | ∆ ∈ End ( ϕ q , Γ)-mod / R (∆),and by assumption, there exists c ∈ L × such that ∇ ′ | ∆ = c · id ∆ . Now wechoose any extension 0 → D → e D → ∆ →
0, and consider the operator ∇ ′ | e D = ∇ β | e D − ∇ β ′ | e D , which is also R -linear and ( ϕ q , Γ)-stable. Since D is F -analytic, we have D ⊂ Ker( ∇ ′ | e D ) and ∇ ′ | e D induces an R -linear and ( ϕ q , Γ)-stable homomorphism ∆ → e D , which is a section of e D → ∆ . Therefore we haveEnd(∆ , D ) = 0 as conclusion. 8et ∆ and D be ( ϕ q , Γ)-modules over R . Then Hom R -mod (∆ , D ) has acanonical ( ϕ q , Γ)-module structure, and we have a natural isomorphismExt( R , Hom R -mod (∆ , D )) ∼ −→ Ext(∆ , D ) . (2)Here, we describe the isomorphism (2) explicitly. Take any extension0 → Hom R -mod (∆ , D ) → e H → R → e h ∈ H of 1 ∈ R . Then both ( ϕ q − e h and ( γ − e h are inHom R -mod (∆ , D ). We put e D = D ⊕ ∆ as an R -module, and define a ( ϕ q , Γ)-action on e D as follows: on D ⊂ e D , we use the original ( ϕ q , Γ)-action. For any x ∈ ∆ and γ ∈ Γ, we define ϕ q | e D ( x ) = ϕ q | ∆ ( x ) + [( ϕ q − e h ]( x ) ,γ | e D ( x ) = γ | ∆ ( x ) + [( γ − e h ]( x ) . Here, for two R -modules V ⊂ V , an element x ∈ V and an operator T which acts on both V and V , the notation T | V i ( x ) means the image of x by T regarding x as an element in V i . Since the isomorphism class [ e D ] of the( ϕ q , Γ)-module e D is independent of the choice of e H and e h , we have a mapExt( R , Hom R -mod (∆ , D )) → Ext(∆ , D ) : [ e H ] [ e D ] , which is the isomorphism (2).Now we present the inverse map of (2). Take any extension 0 → D → e D → ∆ → s : ∆ → e D as an R -module. Note that s is anelement of the ( ϕ q , Γ)-module Hom R -mod (∆ , e D ). Then we have( ϕ q − s, ( γ − s ∈ Hom R -mod (∆ , D )for any γ ∈ Γ. We put e H = Hom R -mod (∆ , D ) ⊕ R e h as an R -module, where R e h is a free R -module of rank 1 of which e h gives a basis. We define a ( ϕ q , Γ)-actionon e H as follows: on Hom R -mod (∆ , D ) ⊂ e H , we use the original ( ϕ q , Γ)-action.We define ϕ q e h = e h + ( ϕ q − s,γ e h = e h + ( γ − s for any γ ∈ Γ . Since the isomorphism class [ e H ] is independent of the choice of e H and s , weobtain a map Ext(∆ , D ) → Ext( R , Hom R -mod (∆ , D )) : [ e D ] [ e H ] , which gives the inverse of (2).For F -analytic ( ϕ q , Γ)-modules D and D over R , Ext an ( D , D ) denotesthe L -subspace of Ext( D , D ) consisting of F -analytic extensions.9 emma 3.2. Let ∆ and D be F -analytic ( ϕ q , Γ) -modules over R . Then theimage of Ext an ( R , Hom R -mod (∆ , D )) by the isomorphism (2) is Ext an (∆ , D ) .Proof. Take any extension 0 → Hom R -mod (∆ , D ) → e H → R →
0, and supposethat, by the isomorphism (2), the class [ e H ] maps to [ e D ] ∈ Ext(∆ , D ). First,we assume that e H is F -analytic. Then we must show that ∆ as a Γ-submoduleof e D is F -analytic. We choose a lift e h of 1 ∈ R in e H as above. Take any γ ∈ Γ sufficiently close to 1. By Lemma 2.11, the element (log γ ) e h is indeed inHom R -mod (∆ , D ). By using Lemma 2.11 again, for any x ∈ ∆, we have(log γ ) | e D ( x ) = (log γ ) | ∆ ( x ) + [(log γ )( e h )]( x ) andlog γ log χ LT ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) e D ( x ) = ∇| ∆ ( x ) + [ ∇ e h ]( x ) . Therefore the operator log γ/ (log χ LT ( γ )) on e D is independent of the choice of γ and e D is F -analytic.Conversely, we will show that the F -analyticity of e D yields that of e H . Takeany section s : ∆ → e D of the projection e D → ∆ as an R -module. Since both ∆and D are F -analytic, the ( ϕ q , Γ)-module Hom R -mod (∆ , D ) is F -analytic. Thus ∇ on Hom R -mod (∆ , D ) is well-defined. By Lemma 2.11, we have(log γ ) e h = (log γ ) s and log γ log χ LT ( γ ) e h = ∇ s. Therefore [log γ/ (log χ LT ( γ ))] e h is independent of the choice of γ and e H is F -analytic.By Lemma 3.2 and [FX13, Cor. 4.4], we have the following: Corollary 3.3.
Let ∆ and D be F -analytic ( ϕ q , Γ) -modules over R . Then Ext an (∆ , D ) is of codimension [ F : Q p ] dim L Hom ( ϕ q , Γ)-mod / R (∆ , D ) in Ext(∆ , D ) . In particular, if Hom ( ϕ q , Γ)-mod / R (∆ , D ) = 0 , then we have Ext an (∆ , D ) = Ext(∆ , D ) . Now we state and prove the main theorem. For a ( ϕ q , Γ)-module D over R ,a finite extension L ′ ⊂ F of L and a character δ : F × → ( L ′ ) × , we denote by D ( δ ) the ( ϕ q , Γ)-module over R L ′ whose underlying R L ′ -module is L ′ ⊗ L D andwhose ( ϕ q , Γ)-action is defined by ϕ q | D ( δ ) ( x ) = δ ( ̟ )(id L ′ ⊗ ϕ q | D )( x ) and γ | D ( δ ) ( x ) = δ ( χ LT ( γ ))(id L ′ ⊗ γ q | D )( x )for any x ∈ D ( δ ) and any γ ∈ Γ. 10 heorem 3.4.
Let D be a ( ϕ q , Γ) -module over R . Suppose that D has a filtra-tion D ⊂ D ⊂ · · · ⊂ D r = D of ( ϕ q , Γ) -modules over R . We put ∆ i = D i /D i − . We assume that the follow-ing conditions hold. (a) . For any ≤ i ≤ r, we have End ( ϕ q , Γ)-mod / R (∆ i ) = L . (b) . For any ≤ i < j ≤ r , we have Hom ( ϕ q , Γ)-mod / R (∆ j , ∆ i ) = 0 . (c) . For any < i ≤ r, the short exact sequence → D i − → D i → ∆ i → does not split.Then there exist a finite extension L ′ of L and a character δ : F × → ( L ′ ) × suchthat, for any ≤ i ≤ r, the ( ϕ q , Γ) -module D i ( δ ) is F -analytic.Proof. We prove it by induction on r . First, we prove the case r = 1. Let D be a( ϕ q , Γ)-module over R such that End ( ϕ q , Γ)-mod / R ( D ) = L . We choose a Z p -basis β , . . . , β n of Lie Γ. For any 1 ≤ i ≤ n , we have ∇ β −∇ β i ∈ End ( ϕ q , Γ)-mod / R ( D ) . Thus there exist c = 0 , c , . . . , c n ∈ L such that ∇ β − ∇ β i = c i · id D for any1 ≤ i ≤ n .Now we choose a Z p -basis γ , . . . , γ n of the free part of Γ such that γ m i i =exp( p m i β i ) for some integers m i ≥
0. We fix a finite extension L ′ of L and acharacter δ : F × → ( L ′ ) × . By Lemma 2.11 , for any x ∈ D ( δ ), we can compute ∇ β i | D ( δ ) ( x ) = (id L ′ ⊗ ∇ β i | D ) ( x ) + log δ ( χ LT ( γ i ))log χ LT ( γ i ) x. Thus, if we find a character δ such that the equality c i + log δ ( χ LT ( γ ))log χ LT ( γ ) − log δ ( χ LT ( γ i ))log χ LT ( γ i ) = 0holds for each 1 ≤ i ≤ n , then D ( δ ) is F -analytic. Actually, we can do it: on thetorsion part of F × and on χ LT ( γ ), put δ = 1. On δ ( χ LT ( γ )) , . . . , δ ( χ LT ( γ n )),put δ ( χ LT ( γ e i i )) = exp ( c i log χ LT ( γ e i i )) , where e i is a sufficiently large integer such that the right-hand side is well-defined. Finally, define δ ( χ LT ( γ i )) as an e i -th root of δ ( χ LT ( γ i )) (hence wemust extend L in general).Next we suppose that the statement for r − ≤ i ≤ r − D i is F -analytic. Then we must show that D r is F -analytic. By the assump-tion (a), (c) and Lemma 3.1, ∆ r is F -analytic. Now we apply the functorHom ( ϕ q , Γ)-mod / R (∆ r , • ) to the short exact sequences0 → ∆ → D → ∆ → , → D → D → ∆ → , ··· → D r − → D r → ∆ r → . ( ϕ q , Γ)-mod / R (∆ r , D i ) =0 for any 1 ≤ i ≤ r −
1. By Corollary 3.3, we have Ext an (∆ r , D r − ) =Ext(∆ r , D r − ) . Therefore D r is F -analytic and we have the conclusion.This theorem, together with Theorem 2.12, implies the following: Corollary 3.5 (Theorem 1.2) . Let V be an overconvergent L -representationof G F such that R ⊗ E † D † ( V ) satisfies the assumptions of Theorem 3.4. Thenthere exist a finite extension L ′ of L and a character δ : G F → ( L ′ ) × such that V ⊗ L L ′ ( δ ) is F -analytic. References [Ber02] L. Berger,
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