aa r X i v : . [ m a t h . N T ] N ov A NOTE ON F -ANALYTIC B -PAIRS by Léo Poyeton
Abstract . —
In this note, we define the notion of F -analytic B -pairs and we prove thatits category is equivalent to the one of F -analytic ( ϕ q , Γ K )-modules. Contents
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Structure of the note. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41. Lubin-Tate extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. Locally, pro-analytic and F -analytic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 53. Rings of periods and ( ϕ q , Γ K )-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84. F -analyticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105. ( B, E )-pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126. A simpler equivalence in the F -analytic case . . . . . . . . . . . . . . . . . . . . . . . . . 19References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Introduction
Let p be a prime and let K be a finite extension of Q p . One of the main tools to study p -adic representations of G K = Gal( Q p /K ) is to operate a “dévissage” of the extension Q p /K through an intermediate extension K ∞ /K which contains most of the ramificationof Q p /K but such that K ∞ /K is nice enough (for example when K ∞ /K is an infinitelyramified p -adic Lie extension).In some sense, the simplest extension one can choose for K ∞ /K is the cyclotomicextension of K . Using the theory of fields of norms [ Win83 ] attached to the cyclo-tomic extension of K , Fontaine has constructed [ Fon90 ] a theory of cyclotomic ( ϕ, Γ K )-modules, which are finite dimensional vector spaces, defined on a local field B K which LÉO POYETON is of dimension 2, and endowed with semilinear actions of a Frobenius ϕ and of Γ K =Gal( K ( µ p ∞ ) /K ) that commute one to another. Moreover, Fontaine has constructed afunctor V D ( V ) which is an equivalence of categories between p -adic representationsof G K and étale ( ϕ, Γ K )-modules (which means that ϕ is of slope 0). The main theorem of[ CC98 ] show that these ( ϕ, Γ K )-modules are overconvergent and it allows us to relate thecyclotomic ( ϕ, Γ K )-modules with classical p -adic Hodge theory, using the fact that theresulting overconvergent ( ϕ, Γ K )-modules give rise to what we still call ( ϕ, Γ K )-modulesbut defined on the cyclotomic Robba ring B † rig ,K .The construction of the p -adic Langlands correspondence for GL ( Q p ) [ Col10 ] reliesheavily on this construction, and in particular on the computations made by Colmez inthe trianguline case [
Col08b ]. A trianguline representation of G Q p of dimension 2 is a2-dimensional p -adic representation of G Q p such that its ( ϕ, Γ)-module is an extension oftwo rank 1 ( ϕ, Γ)-modules. Note that this does not imply that the representation itself isan extension of two rank 1 representations because the ( ϕ, Γ)-modules appearing in this“triangulation” do not need to be étale.In order to extend this correspondence to GL ( F ), it seems necessary to replace thetheory of cyclotomic ( ϕ, Γ K )-modules by Lubin-Tate ( ϕ q , Γ K )-modules, where F ⊂ K and K ∞ /K is generated by the torsion points of a Lubin-Tate group attached to a uniformizerof F . Specializing Fontaine’s constructions, Kisin and Ren have shown that we can attachto each representation of G K a Lubin-Tate ( ϕ q , Γ K )-module D ( V ) over a 2-dimensionallocal field B K (which is not the same as in the cyclotomic case) and such that V D ( V )gives rise to an equivalence of categories when the image is restricted to the subcategoryof étale objects.However, unlike in the cyclotomic case, the resulting Lubin-Tate ( ϕ q , Γ K )-modules areusually not overconvergent. There still exists a subcategory of the one of Lubin-Tate( ϕ q , Γ K )-modules in which the objets are overconvergent, which is the subcategory of F -analytic ( ϕ q , Γ K )-modules. The fact that those are overconvergent is the main theoremof [ Ber16 ]. The generalization of trianguline representations in the Q p -cyclotomic caseto F -analytic representations has been studied in [ FX14 ] (and Kisin and Ren mainlystudied F -analytic crystalline representations in [ KR09 ]).A generalization of trianguline representations in the cyclotomic case for G K has beendone by Nakamura in [ Nak09 ] using the language of Berger’s B -pairs [ Ber08 ] (and theirnatural extension to E -representations which are called E − B − pairs in [ Nak09 ]) but asnoted in the introduction of [
FX14 ], this language does not appear well suited to dealwith Lubin-Tate objects.
NOTE ON F -ANALYTIC B -PAIRS In [
Ber16 , Rem. 10.3] Berger notes that his results and methods should extend toprove that there is an equivalence of categories between F -analytic ( ϕ q , Γ K )-modules and F -analytic B -pairs, and it is this result this note is meant to prove.In the cyclotomic case, it is often useful to switch between cyclotomic ( ϕ, Γ K )-modulesand B -pairs, some properties being easier to prove using one of the categories instead ofthe other, which is why the following theorem might prove useful: Theorem A . —
There is an equivalence of categories between F -analytic B -pairs and F -analytic ( ϕ q , Γ K ) -modules. In particular, a recent preprint from Porat shows that for F -analytic 2-dimensionalrepresentations of G F , V is trianguline in the cyclotomic sense if and only if it is triangulinein the sense of [ FX14 ]. His theorem should extend for F -analytic representations ofarbitrary dimension and it should follow as a straightforward consequence of our theoremA.As stated above, the usual language of B -pairs is not well suited to deal with Lubin-Tate objects and therefore it is quite uneasy to identify the corresponding objects throughthe equivalence of categories of Theorem A. Ding has constructed in [ Din14 ] a variantof Berger’s B -pairs with a Lubin-Tate flavour, which are particularly well suited to dealwith the Lubin-Tate case. For any embedding σ : F → Q p , and for any B -pair D , Dingconstructs what he calls a B σ -pair D σ , such that D D σ is an equivalence of categoriesbetween B -pairs and B σ -pairs. In the F -analytic case, we construct a functor D W ( D )from the category of F -analytic ( ϕ q , Γ K )-modules to the category of F -analytic B id -pairsand which is the natural Lubin-Tate analogue of the constructions of Berger [ Ber08 ]. Inparticular, the following ensues from theorem A but the correspondence between objectsis easier to see:
Theorem B . —
The functor D W ( D ) , from the category of F -analytic ( ϕ q , Γ K ) -modules to the category of F -analytic B id -pairs is an equivalence of categories. Structure of the note
The first three sections of this note are meant to recall the setting, notations and fewproperties of Lubin-Tate extensions, ( ϕ q , Γ K )-modules and locally analytic vectors from[ Ber16 ] that are needed for the rest of this note. In particular, these are pretty muchthe same as [
Ber16 , §1, 2 and 3]. Section 4 explains the notion of F -analyticity in thecase of F -representations and ( ϕ q , Γ K )-modules. In section 5, we recall the notion of LÉO POYETON ( B, E )-pairs, define F -analyticity for ( B, E )-pairs and prove the main theorem of thisnote. In section 6 we explain how to replace the category of F -analytic B -pairs by morenatural objects, following constructions from Yiwen Ding. Acknowledgements
The first version of this note (mainly the first five sections of the current note) waswritten between December 2019 and January 2020. The author would like to thankLaurent Berger and Yiwen Ding for some useful discussions resulting in the first versionof this note. The author would also like to thank Gal Porat for discussing the resultsfrom this note and his remarks and questions which resulted in this new version.
1. Lubin-Tate extensions
Throughout this note, F is a finite extension of Q p , with ring of integers O F , uni-formizer π and residue field k F with cardinal q = p h . We let F = W ( k F )[1 /p ], themaximal unramified extension of Q p inside F , and we let e to be the ramification indexof F . Let Σ be the set of embeddings of F in Q p , and let σ be the absolute Frobenius on F . For τ ∈ Σ, there exists a unique n ( τ ) ∈ { , . . . , h − } such that τ = σ n ( τ ) on k F . Wealso let E to be a field of coefficients which is a finite Galois extension of Q p containing F (hence F Gal ), and write Σ E for Gal( E/ Q p ). We also let Σ = Σ \ { id } .Let S be a formal O F -module Lubin-Tate group law attached to π , such that theendomorphism of multiplication by π is given by the power series [ π ]( T ) = T q + πT . For a ∈ O F , we will denote [ a ]( T ) the power series giving the endomorphism of multiplicationby a for S . Let F n be the field generated by F and the points of π n -torsion, that is theroots of [ π n ]( T ). Let F ∞ = S n ≥ F n , Γ F = Gal( F ∞ /F ) and H F = Gal( F ∞ /F ). Let χ π bethe attached Lubin-Tate character, so that the map g χ π ( g ) induces an isomorphismbetween Γ F and O × F . Note that there exists an unramified character η : G F → Z × p suchthat N F/ Q p ( χ π ) = ηχ cycl , where χ cycl is the cyclotomic character.If K is a finite extension of F , we write K n = KF n and K ∞ = KF ∞ . We let Γ K =Gal( K ∞ /K ) and H K = Gal( K ∞ /K ). We let K η ∞ = Q p ker ηχ cycl , so that K η ∞ ⊂ K ∞ andthat ηχ cycl identifies Gal( K η ∞ /K ) with an open subgroup of Z × p .Now let Γ n = Gal( K ∞ /K n ) so that Γ n = { g ∈ Γ K such that χ π ( g ) ∈ π n O F } . Let u = 0 and for each n ≥
1, let u n ∈ Q p be such that [ π ]( u n ) = u n − , with u = 0. We have v p ( u n ) = 1 /q n − ( q − e for n ≥ F n = F ( u n ). Let Q n ( T ) be the minimal polynomialof u n over F . We have Q ( T ) = T , Q ( T ) = [ π ]( T ) /T and Q n +1 ( T ) = Q n ([ π ]( T )) if n ≥ NOTE ON F -ANALYTIC B -PAIRS Let log LT ( T ) = T + O (deg ≥ ∈ F [[ T ]] denote the Lubin-Tate logarithm map, whichconverges on the open unit disk and satisfies log LT ([ a ]( T )) = a · log LT ( T ) if a ∈ O F . Notethat we have log LT ( T ) = T · Q k ≥ Q k ( T ) /π . We also let exp LT ( T ) denote the inverse oflog LT ( T ).
2. Locally, pro-analytic and F -analytic vectors In this section, we recall the theory of locally analytic vectors of Schneider and Teitel-baum [
ST02 ] but here we follow the constructions of Emerton [
Eme17 ] as in [
Ber16 ].We also define the notion of F -analytic vectors relative to the Galois group of a Lubin-Tate extension, following the definitions of [ Ber16 ]. We will use the following multi-indexnotations: if c = ( c , . . . , c d ) and k = ( k , . . . , k d ) ∈ N d (here N = Z ≥ ), then we let c k = c k · . . . · c k d d .Let G be a p -adic Lie group, and let W be a Q p -Banach representation of G . Let H bean open subgroup of G such that there exists coordinates c , · · · , c d : H → Z p giving riseto an analytic bijection c : H → Z dp . We say that w ∈ W is an H -analytic vector if thereexists a sequence { w k } k ∈ N d such that w k → W and such that g ( w ) = P k ∈ N d c ( g ) k w k for all g ∈ H . We let W H -an be the space of H -analytic vectors. This space injectsinto C an ( H, W ), the space of all analytic functions f : H → W . Note that C an ( H, W )is a Banach space equipped with its usual Banach norm, so that we can endow W H -an with the induced norm, that we will denote by || · || H . With this definition, we have || w || H = sup k ∈ N d || w k || and ( W H -an , || · || H ) is a Banach space.We say that a vector w of W is locally analytic if there exists an open subgroup H as above such that w ∈ W H -an . Let W la be the space of such vectors, so that W la = S H W H -an , where H runs through a sequence of open subgroups of G . The space W la isnaturally endowed with the inductive limit topology, so that it is an LB space.It is often useful to work with a set of “compatible coordinates” of G , that is taking anopen compact subgroup G of G such that there exists local coordinates c : G → ( Z p ) d such that, if G n = G p n − for n ≥
1, then G n is a subgroup of G satisfying c ( G n ) =( p n Z p ) d . By the discussion following [ BC16 , Prop. 2.3], it is always possible to find sucha subgroup G (note however that it is not unique). We then have W la = S n ∈ N W G n -an .In the Lubin-Tate setting, note that the Γ n defined in §1 satisfy this property.In the rest of this note, we will need the following results, which already appear in[ BC16 , §2.1] or [
Ber16 , §2].
LÉO POYETON
Lemma 2.1 . —
Let G and ( G n ) n ∈ N be as in the discussion above. Suppose w ∈ W G n - an . Then for all m ≥ n , we have w ∈ W G m - an and || w || G m ≤ || w || G n . Moreover, wehave || w || G m = || w || when m ≫ .Proof . — This is [ BC16 , Lemm. 2.4].
Lemma 2.2 . —
Let W , X be two Q p -Banach spaces and let π : W → X be a continuouslinear map. If f : G → W is locally analytic, then so is π ◦ f : G → X .Proof . — See [ BC16 , Lemm. 2.2].
Lemma 2.3 . — If W is a ring such that || xy || ≤ || x || · || y || for x, y ∈ W , then1. W H - an is a ring, and || xy || H ≤ || x || H · || y || H if x, y ∈ W H - an ;2. if w ∈ W × ∩ W la , then /w ∈ W la . In particular, if W is a field, then W la is alsoa field.Proof . — See [ BC16 , Lemm. 2.5].Let W be a Fréchet space whose topology is defined by a sequence { p i } i ≥ of seminorms.Let W i be the Hausdorff completion of W at p i , so that W = lim ←− i ≥ W i . The space W la canbe defined but as stated in [ Ber16 ], this space is too small in general for what we areinterested in, and so we make the following definition, following [
Ber16 , Def. 2.3]:
Definition 2.4 . — If W = lim ←− i ≥ W i is a Fréchet representation of G , then we say that avector w ∈ W is pro-analytic if its image π i ( w ) in W i is locally analytic for all i . We let W pa denote the set of all pro-analytic vectors of W .We extend the definition of W la and W pa for LB and LF spaces respectively. Proposition 2.5 . —
Let G be a p -adic Lie group, let B be a Banach G -ring and let W be a free B -module of finite rank, equipped with a compatible G -action. If the B -module W has a basis w , . . . , w d in which g Mat( g ) is a globally analytic function G → GL d ( B ) ⊂ M d ( B ) , then1. W H - an = L dj =1 B H - an · w j if H is a subgroup of G ;2. W la = L dj =1 B la · w j .Let G be a p -adic Lie group, let B be a Fréchet G -ring and let W be a free B -moduleof finite rank, equipped with a compatible G -action. If the B -module W has a basis NOTE ON F -ANALYTIC B -PAIRS w , . . . , w d in which g Mat( g ) is a pro-analytic function G → GL d ( B ) ⊂ M d ( B ) , then W pa = d M j =1 B pa · w j . Proof . — The part for Banach rings is proven in [
BC16 , Prop. 2.3] and the one forFréchet rings is proven in [
Ber16 , Prop. 2.4].The map ℓ : g log p χ π ( g ) gives an F -analytic isomorphism between Γ n and π n O F for n ≫
0. If W is an F -linear Banach representation of Γ K and n ≫
0, then we say,following [
Ber16 ], that an element w ∈ W is F -analytic on Γ n if there exists a sequence { w k } k ≥ of elements of W with π nk w k → g ( w ) = P k ≥ ℓ ( g ) k w k for all g ∈ Γ n .Let W Γ n -an ,F -la denote the space of such elements. Let W F -la = S n ≥ W Γ n -an ,F -la . Lemma 2.6 . —
We have W Γ n - an ,F - la = W Γ n - an ∩ W F - la .Proof . — See [ Ber16 , Lemm. 2.5].Recall that E is a field that contains F Gal . If τ ∈ Σ, then we have the derivative inthe direction τ , which is an element ∇ τ ∈ E ⊗ Q p Lie(Γ F ). It can be constructed in thefollowing way (see also §3.1 of [ DI13 ]). The E -vector space Hom Q p ( F, E ) is generatedby the elements of Σ. If W is an E -linear Banach representation of Γ K and if w ∈ W la and g ∈ Γ K , then there exists elements {∇ τ } τ ∈ Σ of F Gal ⊗ Q p Lie(Γ F ) such that we canwrite log g ( w ) = X τ ∈ Σ τ ( ℓ ( g )) · ∇ τ ( w ) . With the same notation, there exist m ≫ { w k } k ∈ N Σ such that if g ∈ Γ m ,then g ( w ) = P k ∈ N Σ ℓ ( g ) k w k , where ℓ ( g ) k = Q τ ∈ Σ τ ◦ ℓ ( g ) k τ . We have ∇ τ ( w ) = w τ where τ is the Σ-tuple whose entries are 0 except the τ -th one which is 1. If k ∈ N Σ , and ifwe set ∇ k ( w ) = Q τ ∈ Σ ∇ k τ τ ( w ), then w k = ∇ k ( w ) / k !. Remark 2.7 . — If w ∈ W la , then w ∈ W F -la if and only if ∇ τ ( w ) = 0 for all τ ∈ Σ \{ id } . Lemma 2.8 . — If h ∈ Gal( E/ Q p ) acts on E ⊗ Q p Lie(Γ F ) , then h ( ∇ τ ) = ∇ h ◦ τ . Inparticular, ∇ id ∈ F ⊗ Q p Lie(Γ F ) .Proof . — See [ Ber16 , Lemm. 2.8].
Lemma 2.9 . —
Let X and Y be E -representations of Γ n , take g ∈ Gal( E/ Q p ) , and let f : X → Y be a Γ n -equivariant map such that f ( ax ) = g − ( a ) f ( x ) for a ∈ E . If x ∈ X pa ,then ∇ id ( f ( x )) = f ( ∇ g | F ( x )) . LÉO POYETON
3. Rings of periods and ( ϕ q , Γ K ) -modules In this section, we recall the definitions of Lubin-Tate ( ϕ q , Γ K )-modules and of a numberof attached p -adic periods rings.Let F ∞ be the Lubin-Tate extension of F constructed above. Let e E + = { ( x , x , . . . ),with x n ∈ O C p /π and such that x qn +1 = x n for all n ≥ } . This ring is endowed with thevaluation v E ( · ) defined by v E ( x ) = lim n → + ∞ q n v p ( b x n ) where b x n ∈ O C p lifts x n . The ring e E + is complete for v E ( · ). If the { u n } n ≥ are as in section 1, then u = ( u , u , . . . ) ∈ e E + and v E ( u ) = q/ ( q − e . Let e E be the fraction field of e E + .Let W F ( · ) = O F ⊗ O F W ( · ) be the F -Witt vectors. Let f A + = W F ( e E + ), f A = W F ( e E )and let e B + = f A + [1 /π ] and e B = f A [1 /π ]. These rings are preserved by the Frobenius map ϕ q = id ⊗ ϕ h . By [ Col02 , §9.2], there exists u ∈ f A + , whose image in e E + is u , and suchthat ϕ q ( u ) = [ π ]( u ) and g ( u ) = [ χ π ( g )]( u ) if g ∈ Γ F .Every element of e B + [1 / [ u ]] can be written uniquely as a sum P k ≫−∞ π k [ x k ] where { x k } k ∈ Z is a bounded sequence of e E . If r ≥
0, we define a valuation V ( · , r ) on e B + [1 / [ u ]]by V ( x, r ) = inf k ∈ Z ke + p − pr v E ( x k ) ! if x = X k ≫−∞ π k [ x k ] . If I is a closed subinterval of [0; + ∞ [, then let V ( x, I ) = inf r ∈ I V ( x, r ). The ring e B I isdefined to be the completion of e B + [1 / [ u ]] for the valuation V ( · , I ) if 0 / ∈ I . If I = [0; r ],then e B I is the completion of e B + for V ( · , I ). Let f A I be the ring of integers of e B I for V ( · , I ).For ρ >
0, let ρ ′ = ρ · e · p/ ( p − · ( q − /q . We have V ( u i , r ) = i/r ′ for i ∈ Z if r > Ber16 , §3]).Let I be either a subinterval of ]1; + ∞ [ or such that 0 ∈ I , and let f ( Y ) = P k ∈ Z a k Y k be a power series with a k ∈ F and such that v p ( a k ) + k/ρ ′ → + ∞ when | k | → + ∞ for all ρ ∈ I . The series f ( u ) converges in e B I and we let B IF denote the set of f ( u ) where f ( Y )is as above. It is a subring of e B IF = ( e B I ) H F , which is stable under the action of Γ F . TheFrobenius map gives rise to a map ϕ q : B IF → B qIF . If m ≥
0, then ϕ − mq ( B q m IF ) ⊂ e B IF andwe let B IF,m = ϕ − mq ( B q m IF ) so that B IF,m ⊂ B IF,m +1 for all m ≥ B † ,r rig ,F denote the ring B [ r ;+ ∞ [ F . This is a subring of e B [ r ; s ] F for all s ≥ r . Let B † ,rF denote the set of f ( u ) ∈ B † ,r rig ,F such that in addition { a k } k ∈ Z is a bounded sequence.Let B † F = ∪ r ≫ B † ,rF . This a Henselian field (cf. §2 of [ M + ]), whose residue field E F is isomorphic to F q (( u )). Let K be a finite extension of F . By the theory of the fieldof norms (see [ Win83 ]), there corresponds to
K/F a separable extension E K / E F , ofdegree [ K ∞ : F ∞ ]. Since B † F is a Henselian field, there exists a finite unramified extension NOTE ON F -ANALYTIC B -PAIRS B † K / B † F of degree f = [ K ∞ : F ∞ ] whose residue field is E K (cf. §3 of [ M + ]). There existtherefore r ( K ) > x , . . . , x f in B † ,r ( K ) K such that B † ,sK = ⊕ fi =1 B † ,sF · x i forall s ≥ r ( K ). Note that the rings B † K are actually contained inside e B . If r ( K ) ≤ min( I ),then let B IK denote the completion of B † ,r ( K ) K for V ( · , I ), so that B IK = ⊕ fi =1 B IF · x i . Let B IK,m = ϕ − mq ( B q m IK ) and B IK, ∞ = ∪ m ≥ B IK,m so that B IK,m ⊂ e B IK = ( e B I ) H K . We also let B K to be the p -adic completion of B † K inside e B , and A K its ring of integers for the p -adictopology (note that we could have defined A F as the p -adic completion of O F [[ u ]][1 /u ]inside f A , put B F = A F [1 /π ] and used the same argument as in the beginning of [ Col08a ,§6.1] to define B K ). Let B be the p -adic completion of S K/F B K inside e B .Let B † ,r rig ,K denote the Fréchet completion of B † ,rK for the valuations { V ( · , [ r ; s ]) } s ≥ r .Let B † ,r rig ,K,m = ϕ − mq ( B † ,q m r rig ,K ) and B † ,r rig ,K, ∞ = ∪ m ≥ B † ,r rig ,K,m . We have B † ,r rig ,K, ∞ ⊂ e B [ r ; s ] K for all s ≥ r . Let e B † ,r rig denote the Fréchet completion of e B + [1 / [ u ]] for the valuations { V ( · , [ r ; s ]) } s ≥ r ; e B † ,r rig is a subring of e B [ r ; s ] for all s ≥ r . Let e B † rig = ∪ r ≫ e B † ,r rig and e B † ,r rig ,K =( e B † ,r rig ) H K and e B † rig ,K = ( e B † rig ) H K . Theorem 3.1 . —
Let I = [ r ℓ ; r k ] with ℓ ≤ k , let K be a finite extension of F , and let m ≥ be such that t π and t π /Q k belong to ( e B IF ) Γ m + k - an ,F - la .1. ( e B IF ) Γ m + k - an ,F - la ⊂ B IF,m ;2. ( e B IK ) F - la = B IK, ∞ ;3. ( e B † ,r ℓ rig ,K ) F - pa = B † ,r ℓ rig ,K, ∞ .Proof . — This is [ Ber16 , Thm. 4.4].Recall that K η ∞ /K is the extension of K attached to ηχ cycl . Let Γ ′ K = Gal( K η ∞ /K ). Let B † K,η , B IK,η and B † rig ,K,η be as in [ Ber16 , §8]. By the same arguments as in [
Ber16 , §8],there is an equivalence of categories between étale ( ϕ, Γ ′ K )-modules over E ⊗ Q p B † rig ,K,η (it is also true over E ⊗ Q p B † K,η ) and E -representations of G K . We will also denote by e B † rig ,η the ring e B † rig in the specific case of F = Q p , so that e B † rig = F ⊗ F e B † rig ,η . Note thatthe ring e B † rig ,η does actually not depend on η but we use this notation for convenience.A ( ϕ q , Γ K )-module over B K is a B K -vector space D of finite dimension d , along with asemilinear Frobenius map ϕ q and a commuting continuous and semilinear action of Γ K .We say that D is étale if there exists a basis of D in which Mat( ϕ ) belongs to GL d ( A K ).By specializing the constructions of [ Fon90 ], Kisin and Ren prove the following theorem[
KR09 , Thm. 1.6]. LÉO POYETON
Theorem 3.2 . —
The functors V ( B ⊗ F V ) H K and D ( B ⊗ B K D ) ϕ q =1 give riseto mutually inverse equivalences of categories between the category of F -linear represen-tations of G K and the category of étale ( ϕ q , Γ K ) -modules over B K . We say that a ( ϕ q , Γ K )-module D is overconvergent if there exists a basis of D in whichthe matrices of ϕ q and of all g ∈ Γ K have entries in B † K . This basis generates a B † K -vectorspace D † which is canonically attached to D . Recall that E is a field of coefficients thatis a finite extension of F Gal . Theorem 3.2 extends more generally to an equivalence ofcategories between the category of E -linear representations of G K and the category ofétale ( ϕ q , Γ K )-modules over E ⊗ F B K .The main result of [ CC98 ] states that if F = Q p , then every étale ( ϕ q , Γ K )-moduleover B K is overconvergent but if F = Q p , then this is no longer the case (see [ FX14 ] forexample).
Lemma 3.3 . — If R denotes one of the rings B † ,r rig ,K , B † rig ,K , e B † ,r rig ,K or e B † rig ,K , and if M is a sub- R -module of a free R -module of finite type, then the following are equivalent:1. M is free ;2. M is closed ;3. M is finitely generated.Proof . — See [ Ber08 , Prop. 1.1.1]. F -analyticity We now explain what it means for a representation or a ( ϕ q , Γ K )-module to be F -analytic, and we give a few related properties.We say, following [ Ber16 , §7] that an F -linear representation V of G K is F -analytic if C p ⊗ τF V is the trivial C p -semilinear representation of G K for all embeddings τ = id ∈ Σ.The following lemma shows that the condition for an E -representation to be F -analyticdepends only on the restriction of the elements of Σ E to F . Lemma 4.1 . — If V is an E -representation of G K , then the following are equivalent:1. V seen as an F -representation is F -analytic;2. C p ⊗ gE V is the trivial C p -semilinear representation of G K for all g ∈ Gal( E/ Q p ) such that g | F = id .Proof . — See [ Ber16 , Lemm. 7.2].
NOTE ON F -ANALYTIC B -PAIRS If D † rig is a ( ϕ q , Γ K )-module over B † rig ,K , and if g ∈ Γ K is close enough to 1, then theseries log( g ) = log(1 + ( g − ∇ g : D † rig → D † rig . Themap Lie Γ K → End( D † rig ) arising from v
7→ ∇ exp( v ) is Q p -linear, and we say, following[ KR09 , §2.1], [
FX14 , §1.3] and [
Ber16 , §7], that D † rig is F -analytic if this map is F -linear. This is the same as asking the elements of D † rig to be pro- F -analytic vectors forthe action of Γ K .Berger proved [ Ber16 , Thm. 10.1] that the Lubin-Tate ( ϕ q , Γ K )-modules attachedto F -analytic representations are overconvergent. He showed how to attach to an E -representation V of G K that is F -analytic an étale ( ϕ q , Γ K )-module D † rig ( V ) over B † rig ,K which is F -analytic and proved [ Ber16 , Thm. 10.4] that the functor V D † rig ( V ) givesrise to an equivalence of categories between the category of F -analytic E -representationsof G K and the category of étale F -analytic Lubin-Tate ( ϕ q , Γ K )-modules over E ⊗ F B † rig ,K .We will give an equivalent condition for a Lubin-Tate ( ϕ q , Γ K )-module over E ⊗ F B † rig ,K to be F -analytic but first we recall a few of Berger’s results about locally analytic periodsinside some of the rings we have defined.Given τ ∈ Σ and f ( Y ) = P k ∈ Z a k Y k with a k ∈ F , let f τ ( Y ) = P k ∈ Z τ ( a k ) Y k . For τ ∈ Σ, let e n ( τ ) be the lift of n ( τ ) ∈ Z /h Z belonging to { , . . . , h − } . Recall that E is a finiteextension of F that contains F Gal and that if τ ∈ Σ, then we have ∇ τ ∈ E ⊗ F Lie(Γ F ).The field E is a field of coefficients, so that G K acts E -linearly below.Let t π = log LT ( u ) ∈ B +rig ,K . Note that we actually have t π ∈ B +rig ,F , and that ϕ q ( t π ) = πt π and g ( t π ) = χ π ( g ) t π if g ∈ G F . Let y τ = ( τ ⊗ ϕ e n ( τ ) )( u ) ∈ O E ⊗ O F f A + . We have g ( y τ ) = [ χ π ( g )] τ ( y τ ) and ϕ q ( y τ ) = [ π ] τ ( y τ ) = τ ( π ) y τ + y qτ . Let t τ = ( τ ⊗ ϕ e n ( τ ) )( t π ) =log τ LT ( y τ ).We have ∇ τ ( y τ ) = t τ · v τ where v τ = ( ∂ ( T ⊕ LT U ) /∂U ) τ ( y τ ,
0) is a unit (see §2.1 of[
KR09 ]). Let ∂ τ = t − τ v − τ ∇ τ so that ∂ τ ( y τ ) = 1. If τ, υ ∈ Σ, then ∂ τ ◦ ∂ υ = ∂ υ ◦ ∂ τ , and ∂ τ ( y υ ) = 0 if τ = υ . Lemma 4.2 . —
We have ∂ τ (( E ⊗ F e B † rig ,K ) pa ) ⊂ ( E ⊗ F e B † rig ,K ) pa .Proof . — See [ Ber16 , Lemm. 5.2].
Proposition 4.3 . —
Let M be a ( ϕ q , Γ K ) -module over E ⊗ F ( e B † rig ,K ) pa . Let Sol( M ) = { x ∈ M such that ∇ τ ( x ) = 0 for all τ ∈ Σ } . If for all τ ∈ Σ , ∇ τ ( M ) ⊂ t τ · M , then there exists a unique ( ϕ q , Γ K ) -module D † rig over E ⊗ F B † rig ,K such that Sol( M ) = ( E ⊗ F ( e B † rig ,K ) F - pa ) ⊗ E ⊗ F e B † rig ,K D † rig and such that M = ( E ⊗ F ( e B † rig ,K ) pa ) ⊗ E ⊗ F B † rig ,K D † rig , and D † rig is an F -analytic ( ϕ q , Γ K ) -module. LÉO POYETON
Moreover, if D is a ( ϕ q , Γ K ) -module over E ⊗ F B † rig ,K , and if M = ( E ⊗ F e B † rig ,K ) ⊗ E ⊗ F B † rig ,K D , then D is F -analytic if and only if for all τ ∈ Σ , ∇ τ ( M pa ) ⊂ t τ · M pa .Proof . — We first prove the first part of the theorem. Let M be a ( ϕ q , Γ K )-module over E ⊗ F ( e B † rig ,K ) pa . Theorem 6.1 of [ Ber16 ] shows thatSol( M ) = { x ∈ M such that ∇ τ ( x ) = 0 for all τ ∈ Σ } is a free E ⊗ F ( e B † rig ,K ) F -pa -module of rank d such that( E ⊗ F e B † rig ,K ) ⊗ E ⊗ F ( e B † rig ,K ) F -pa ) F -pa Sol( M ) = ( E ⊗ F e B † rig ,K ) ⊗ E D . By (3) of theorem 3.1, we have ( e B † rig ,K ) F -pa = B † rig ,K, ∞ = S n ≥ B † rig ,K,n . Since Γ K istopologically of finite type, there exist n ≥
0, and a basis s , . . . , s d of Sol( M ) such thatMat( ϕ q ) ∈ GL d ( E ⊗ F B † rig ,K,n ) and Mat( g ) ∈ GL d ( E ⊗ F B † rig ,K,n ) for all g ∈ Γ K . If D † rig = ⊕ di =1 ( E ⊗ F B † rig ,K ) · ϕ nq ( s i ), then D † rig is a ( ϕ q , Γ K )-module over E ⊗ F B † rig ,K suchthat Sol( M ) = ( E ⊗ F ( e B † rig ,K ) F -pa ) ⊗ E ⊗ F B † rig ,K D † rig .The module D † rig is uniquely determined by this condition: if there are two such modulesand if X denotes the change of basis matrix and P , P denote the matrices of ϕ q , then X ∈ GL d ( E ⊗ F B † rig ,K,n ) for n ≫
0, and the equation X = P − ϕ ( X ) P implies that X ∈ GL d ( E ⊗ F B † rig ,K ).Since Sol( M ) is a free E ⊗ F ( e B † rig ,K ) F -pa -module, D † rig is also free of the same rank.Now, let D be a ( ϕ q , Γ K )-module over E ⊗ F B † rig ,K , such that M = ( E ⊗ F e B † rig ,K ) pa ⊗ E ⊗ F B † rig ,K D is such that for all τ ∈ Σ , ∇ τ ( M ) ⊂ t τ · M . We thenhave D ⊂ Sol( M ) so that D is F -analytic by the above. If D is an F -analytic ( ϕ q , Γ K )-module over E ⊗ F B † rig ,K , then we have ∇ τ ( x ) = 0 for all x ∈ D by remark 2.7 and so ∇ τ ( M ) ⊂ t τ · M for M = ( E ⊗ F e B † rig ,K ) pa ⊗ E ⊗ F B † rig ,K D by lemma 4.2. ( B, E ) -pairs In this section, we quickly recall the definitions of (
B, E )-pairs and define the notionof F -analytic ( B, E )-pairs.Let B +dR , B dR , B +cris and B cris be the usual Fontaine’s rings of p -adic periods, definedfor example in [ Fon94 ]. These rings come equipped with an action of G Q p , and the rings B +cris and B cris are endowed with an injective Frobenius ϕ . We let B e = ( B cris ) ϕ =1 . Bergerdefined in [ Ber08 ] the notion of B -pairs, that is pairs W = ( W e , W + dR ), where W e is a free B e -module of finite rank, equipped with a semilinear continuous action of G K and where W + dR is a G K -stable B +dR -lattice inside W dR = B dR ⊗ B e W e . To a p -adic representation V , NOTE ON F -ANALYTIC B -PAIRS one can attach the B -pair W ( V ) = ( B e ⊗ Q p V, B +dR ⊗ Q p V ), and the functor V W ( V )is fully faithful since B e ∩ B +dR = Q p . Recall that t is the usual t in p -adic Hodge theory(note that t corresponds to the element t p for F = Q p ) and that B +dR /t B +dR = C p .Berger showed [ Ber08 , Thm. 2.2.7] how to attach to any B -pair a cyclotomic ( ϕ, Γ)-module D ( W ) on the (cyclotomic) Robba ring, and that this functor induces an equiva-lence of categories.Let E be a field of coefficients as previously. Let B e,E = E ⊗ Q p B e , B +dR ,E = E ⊗ Q p B +dR and B dR ,E = E ⊗ Q p B dR , where G Q p acts E -linearly on E . A ( B, E )-pair is apair W = ( W e , W + dR ), where W e is a free B e,E -module of finite rank, equipped with asemilinear continuous action of G K and where W + dR is a G K -stable B +dR ,E -lattice inside W dR = B dR ,E ⊗ B e,E W e . To an E representation V , one can attach the ( B, E )-pair W ( V ) = ( B e ⊗ Q p V, B +dR ⊗ Q p V ), and this functor is once again fully faithful. Theorem2.2.7 of [ Ber08 ] has been extended by Nakamura [
Nak09 , Thm. 1.36] for (
B, E )-pairsand cyclotomic E -( ϕ, Γ)-modules, that is ( ϕ, Γ)-modules over the cyclotomic Robba ringtensored by E over Q p .Let F, E be as in §1. Note that we have an isomorphism E ⊗ Q p F ≃ Q τ ∈ Σ E , given by a ⊗ b ( aτ ( b )) τ ∈ Σ . Since F ⊂ B +dR , we have natural isomorphisms E ⊗ Q p B +dR ≃ ( E ⊗ Q p F ) ⊗ F B +dR ≃ ( Y τ ∈ Σ E ) ⊗ F B +dR ≃ Y τ ∈ Σ B +dR ,τ where B +dR ,τ = E ⊗ τF B +dR , and E ⊗ Q p B dR ≃ Y τ ∈ Σ B dR ,τ where B dR ,τ = E ⊗ τF B dR .Using these decompositions, we get decompositions W + dR ≃ Q τ ∈ Σ W + dR,τ and W dR ≃ Q τ ∈ Σ W dR,τ .We say that a ( B, E )-pair is F -analytic if for all τ ∈ Σ , W + dR,τ /tW + dR,τ is the trivial C p -semilinear representation of G K . The following lemma shows that this definition iscompatible with the one of F -analytic representation: Lemma 5.1 . —
Let V be an E -representation of G K . Then V is F -analytic if and onlyif the ( B, E ) -pair W ( V ) = ( W e , W +dR ) = ( B e ⊗ Q p V, B +dR ⊗ Q p V ) is F -analytic.Proof . — We have B +dR /t B +dR = C p , so that W + dR /tW + dR = C p ⊗ Q p V ≃ Q τ ∈ Σ ( C p ⊗ τF V ),and W + dR,τ /tW + dR,τ = C p ⊗ τF V , and so the equivalence is clear. Lemma 5.2 . —
We have B e,E = E ⊗ F ( e B † rig [1 /t ]) ϕ q =1 . LÉO POYETON
Proof . — First, recall that B e = ( e B † rig ,η [1 /t ]) ϕ =1 (this is [ Ber08 , Lemm. 1.1.7]). Since ϕ q is F -linear, we have ( e B † rig [1 /t ]) ϕ q =1 = ( F ⊗ F e B † rig ,η [1 /t ]) ϕ q =1 = F ⊗ F ( e B † rig ,η [1 /t ]) ϕ h =1 .Now since Gal( F / Q p ) acts F -semi-linearly on ( e B † rig ,η [1 /t ]) ϕ h =1 by ϕ , Speiser’s lemmaimplies that ( e B † rig ,η [1 /t ]) ϕ h =1 = F ⊗ Q p B e . Thus, we get that B e,E = E ⊗ Q p B e = E ⊗ F F ⊗ F ( F ⊗ B e )and what we just did implies that B e,E = E ⊗ F ( e B † rig [1 /t ]) ϕ q =1 , which is what we wanted.Note that this is basically the same proof (tensored by E over F ) as [ BDM19 , Prop.2.2] but using e B † rig [1 /t ] instead of F ⊗ F B cris . Lemma 5.3 . —
1. The t -adic valuation of the τ ′ -component of the image of t τ by the map e B +rig → F ⊗ Q p B dR = Q τ ′ ∈ Σ B dR given by x
7→ { ( τ ′ ⊗ ϕ n ( τ ′ ) )( x ) } τ ′ ∈ Σ is if τ ′ = τ − and otherwise.2. There exists u ∈ ( F ⊗ d Q p unr ) × such that Q τ ∈ Σ t τ = u · t in e B +rig .Proof . — These are items 2 and 3 of [ BDM19 , Prop. 2.4], using e B +rig instead of F ⊗ F B +cris .We now explain how to attach F -analytic ( B, E )-pairs to F -analytic ( ϕ q , Γ K )-modulesover ( E ⊗ F B † rig ,K ) and vice-versa.The previous lemma allows us to see E ⊗ F e B † rig [1 /t ] as a B e,E -module.Let Ω = { ( τ, n ) ∈ Gal( E/ Q p ) × Z such that n ( τ | F ) ≡ n mod h } . For n ≥
0, let r n = p n − ( p − r >
0, let n ( r ) be the least integer n such that r n ≥ r . For r ≥
0, we let Ω r = { ( τ, n ) ∈ Ω such that n ≥ n ( r ) } . For g = ( τ, n ) ∈ Ω, we let τ ( g ) = τ and n ( g ) = n . If min( I ) ≥ r and if g ∈ Ω r , we have a map ι g : E ⊗ F e B I → E ⊗ τ ( g ) | F F B +dR = B dR ,τ ( g ) | F , defined in [ Ber16 , §5] and given by x ( g − ⊗ ( g | − F ⊗ ϕ − n ( g ) ))( x ). Lemma 5.4 . —
Let W be a ( B, E ) -pair of rank d , and let f D r ( W ) = n y ∈ ( E ⊗ F e B † ,r rig [1 /t ]) ⊗ B e,E W e such that ι g ( y ) ∈ W + dR,τ ( g ) | F for all g ∈ Ω r o . Then:1. f D r ( W ) is a free E ⊗ F e B † ,r rig -module of rank d ;2. f D r ( W )[1 /t ] = ( E ⊗ F e B † ,r rig [1 /t ]) ⊗ B e,E W e . NOTE ON F -ANALYTIC B -PAIRS Proof . — The main ideas of the proof are basically the same as in [
Ber08 , Lemm. 2.2.1].Let τ ∈ Σ, let r >
0, let g = ( τ, n ( r )) and let f D rn ( r ) ,τ ( W ) = n y ∈ ( E ⊗ F e B † ,r rig [1 /t ]) ⊗ B e,E W e such that ι g ( y ) ∈ W + dR,τ o . It is a free E ⊗ F e B † ,r rig -module of rank d , since it is generated by d elements generatinga basis of W + dR,τ . Let f D rτ ( W ) = n y ∈ ( E ⊗ F e B † ,r rig [1 /t ]) ⊗ B e,E W e such that ι g ( y ) ∈ W + dR,τ for all g ∈ Ω r with τ ( g ) = τ o . Note that f D rτ ( W ) is closed in f D rn ( r ) ,τ ( W ). Now let f D rτ ( W ) = T τ ∈ Σ f D rτ ( W ). As a finiteintersection of closed subsets of f D rn ( r ) , id ( W ), it is closed in f D rn ( r ) , id ( W ) so that it is free ofrank ≤ d as a E ⊗ F e B † ,r rig -module by lemma 3.3.Now let x ∈ ( E ⊗ F e B † ,r rig [1 /t ]) ⊗ B e,E W e , and let us prove that there exists m ≥ t m x ∈ f D r ( W ), which will imply items 1 and 2.For τ ∈ Σ, let e , . . . , e d be a basis of W e . Since ϕ q ( e i ) = e i , there exists m τ such that,for all g ∈ Ω r such that τ ( g ) = τ , the image by ι g of the t m τ τ e i belong to W + dR . Sincefor all τ ∈ Σ, t τ ∈ e B +rig , and since the product Q τ ∈ Σ t τ ∈ t · ( e B +rig ) × by lemma 5.3, thisimplies that, for m = max( m τ ) and for all τ ∈ Σ, the image of t m e i belong to W + dR .Now since B e,E = S j ≥ E ⊗ F ( e B +rig ) ϕ q = q j t − j , there exists m such that t m x belongs tothe e B † ,r rig -module generated by the e i , so that m = m + m satisfies the condition wewanted.If W is a ( B, E )-pair, we let f D ( W ) = ( E ⊗ F e B † rig ) ⊗ E ⊗ F e B † ,r rig f D r ( W ), and if I is asubinterval of [ r ; + ∞ [, we let f D I ( W ) = ( E ⊗ F e B I ) ⊗ E ⊗ F e B † ,r rig f D r ( W ). By the sameargument as in [ Ber08 , Lemm. 2.2.2], this does not depend on the choice of r ∈ I . Proposition 5.5 . — If W is a ( B, E ) -pair of rank d , then there exists a unique ( ϕ q , Γ ′ K ) -module D η ( W ) over E ⊗ F B † rig ,K,η such that ( E ⊗ F e B † rig ) ⊗ E ⊗ F B † rig ,K,η D η ( W ) = f D ( W ) .Proof . — This is [ Ber08 , Prop. 2.2.5] up to a tensor product, and using the twistedcyclotomic case instead of the classical one, but again by using [
Ber16 , §8], it does notchange the arguments of the proof.For r ≥ D η ( W ) and all its structures are defined over E ⊗ F B † ,r rig ,K,η ,we let D rη ( W ) be the associated ( E ⊗ F B † ,r rig ,K,η )-module so that D η ( W ) = ( E ⊗ F B † rig ,K,η ) ⊗ E ⊗ F B † ,r rig ,K,η D rη ( W ). For I = [ r ; s ], we let D Iη = ( E ⊗ F B IK,η ) ⊗ E ⊗ F B † ,r rig ,K,η D rη ( W ). Let f D IK ( W ) = ( f D I ( W )) H K and f D K ( W ) = f D ( W ) H K , so that f D IK ( W ) =( E ⊗ F e B IK ) ⊗ E ⊗ F B IK,η D Iη ( W ) and f D K ( W ) = ( E ⊗ F e B † rig ,K ) ⊗ E ⊗ F B † rig ,K,η D η ( W ) (since D η ( W ) is invariant under H K ). LÉO POYETON
Proposition 5.6 . —
We have1. f D IK ( W ) la = ( E ⊗ F e B IK ) la ⊗ E ⊗ F B IK,η D Iη ( W ) ;2. f D K ( W ) pa = ( E ⊗ F e B † rig ,K ) pa ⊗ E ⊗ F B † rig ,K,η D η ( W ) .Proof . — The same proof as [ KR09 , §2.1] shows that the elements of D Iη ( W ) are locallyanalytic vectors, and the result now follows from proposition 2.5. Theorem 5.7 . — If W is an F -analytic ( B, E ) -pair of rank d , then there exists a unique F -analytic ( ϕ q , Γ K ) -module D ( W ) over E ⊗ F B † rig ,K such that ( E ⊗ F e B † rig ) ⊗ E ⊗ F B † rig ,K D ( W ) = f D ( W ) .Proof . — Let W be an F -analytic ( B, E )-pair of rank d , and let f D K ( W ) be as above.Let r ≥ y ∈ ( f D K ( W ) r ) pa . Let τ ∈ Σ \ { id } and letΩ τ,r = { g ∈ Ω such that n ( g ) ≥ n ( r ) and τ ( g ) = τ } . Let g ∈ Ω τ,r . We have ι g ( y ) ∈ W + dR,τ . Write x g for the image of ι g ( y ) in W + dR,τ /tW + dR,τ .Since the filtration on W dR,τ is Galois stable, we get that x g is invariant under H K (since ι g ( y ) is), and is a locally analytic vector of ( W + dR,τ /tW + dR,τ ) H K using the fact that y ∈ ( f D K ( W ) r ) pa and proposition 2.2. Note that ∇ id = 0 on (( W + dR,τ /tW + dR,τ ) H K ) la since W is F -analytic and by [ Ber16 , Prop. 2.10]. This shows that ∇ id ( x g ) = 0 and so ∇ id ( ι g ( y )) = 0 mod t π (recall that t and t π both generate the kernel of θ in B +dR bylemma 5.3). Using the fact that ι g ◦ ∇ τ = ∇ id ◦ ι g by lemma 2.9, this implies that t π | ι g ◦ ∇ τ ( y ) in W + dR,τ . By lemma 5.3, this proves that ι g (( Q τn ) − · ∇ τ ( y )) ∈ W + dR,τ for n = n ( g ). By definition of f D r ( W ), this proves that ∇ τ ( y ) ∈ Q τn · f D r ( W ) for all n ≥ n ( r ),and so ∇ τ is divisible by + ∞ Q n = n ( r ) Q τn in f D r ( W ) (the argument for the divisibility by aninfinite product is the same as the one given in the proof of [ Ber16 , Lemm. 10.2]), henceby t τ .In particular, for all τ ∈ Σ \ { id } , we have ∇ τ ( f D r ( W ) pa ) ⊂ t τ · f D r ( W ) pa . Byproposition 4.3, there exists a unique ( ϕ q , Γ K )-module D † rig over E ⊗ F B † rig ,K such that( E ⊗ F e B † rig ) ⊗ E ⊗ F B † rig ,K D † rig = f D ( W ), which is what we wanted.We now explain how to attach an F -analytic ( B, E )-pair to an F -analytic ( ϕ q , Γ K )-module. Proposition 5.8 . — If D is a ϕ q -module over B † rig ,K , then there exists r ( D ) ≥ r ( K ) such that, for all r ≥ r ( D ) , there exists a unique sub B † ,r rig ,K -module D r of D such that:1. D = B † rig ,K ⊗ B † ,r rig ,K D r ; NOTE ON F -ANALYTIC B -PAIRS
2. the B † ,qr rig ,K -module B † ,qr rig ,K ⊗ B † rig ,K ,r D r has a basis contained inside ϕ q ( D ) . Moreover,if D is a ( ϕ q , Γ K ) -module, one has g ( D r ) = D r for all g ∈ Γ K .Proof . — This is exactly the same proof as [ Ber08 , Thm. I.3.3] but using Lubin-Tate( ϕ q , Γ K )-modules instead of cyclotomic ones, and tensoring by E over F . Proposition 5.9 . — If D is a ( ϕ q , Γ K ) -module over E ⊗ F B † rig ,K , free of rank d , then1. W e ( D ) = ( E ⊗ F e B † rig ,K [1 /t ] ⊗ B † rig ,K D ) ϕ q =1 is a free B e,E -module of rank d which is G K -stable;2. W + dR = Q τ ∈ Σ ( E ⊗ F B +dR ) ⊗ ι g E ⊗ F B † ,rn ( g )rig ,K D r n ( g ) ! g ∈ Ω r,τ does not depend on n ( g ) ≫ and is a free E ⊗ Q p B +dR = ( B +dR τ ) τ ∈ Σ -module of rank d and G K -stable.3. W ( D ) = ( W e ( D ) , W + dR ( D )) is a ( B, E ) -pair. Moreover, if D is F -analytic, then sois W ( D ) .Proof . — The proof of items 1 and 2 is the same as [ Ber08 , Prop. 2.2.6]. Assume nowthat D is F -analytic, and let us prove that W ( D ) is F -analytic. Let τ ∈ Σ \ { id } .By item 2, we have W + dR,τ = ( E ⊗ F B +dR ) ⊗ ι g E ⊗ F B † ,rn ( g )rig ,K D r n ( g ) for some g ∈ Ω r,τ . Wecan find a basis e , . . . , e d of D r n ( g ) over E ⊗ F B † ,r n ( g ) rig ,K such that the image of the basis ι g ( e ) , . . . , ι g ( e d ) of W + dR,τ over E ⊗ F B +dR modulo t π is a basis of the E ⊗ F C p -representation W + dR,τ /tW + dR,τ .Since the e i are pro-analytic vectors of D r n ( g ) for the action of Γ K , the same argumentas in the proof of theorem 5.7 shows that their image in W + dR,τ /tW + dR,τ are invariant under H K and locally analytic vectors of ( W + dR,τ /tW + dR,τ ) H K . Since ∇ τ ( E ⊗ F e B † ,r n ( g ) rig ,K ) pa ⊗ E ⊗ F B † ,rn ( g )rig ,K D r n ( g ) ! ⊂ t τ · ( E ⊗ F e B † ,r n ( g ) rig ,K ) pa ⊗ E ⊗ F B † ,rn ( g )rig ,K D r n ( g ) ! by lemma 2.7 and since W + dR,τ = ( E ⊗ F B +dR ) ⊗ ι g E ⊗ F B † ,rn ( g )rig ,K (( E ⊗ F e B † ,r n ( g ) rig ,K ) pa ⊗ E ⊗ F B † ,rn ( g )rig ,K D r n ( g ) )we get that ∇ id ( e i ) = 0 mod t π for all i since ι g ◦ ∇ τ = ∇ id ◦ ι g by lemma 2.9 and since ι g ( t τ ) = t π .This implies that ∇ id = 0 on ( W + dR,τ /tW + dR,τ ) H K , la so that ( W + dR,τ /tW + dR,τ ) is C p -admissible as a E ⊗ F C p representation of G K , using the discussion following [ BC16 ,Thm. 4.11]. LÉO POYETON
Theorem 5.10 . —
The two functors W D ( W ) and D W ( D ) are inverse oneto another and induce an equivalence of categories between the category of F -analytic ( B, E ) -pairs and the category of F -analytic ( ϕ q , Γ K ) -modules.Proof . — Let W = ( W e , W + dR ) be an F -analytic ( B, E )-pair and let D = D ( W ). Bydefinition of W ( D ), we have( E ⊗ F e B † rig [1 /t ]) ⊗ B e,E W e ( D ) = ( E ⊗ F e B † rig [1 /t ]) ⊗ E ⊗ F B † rig ,K D and by definition of D ( W ), we have( E ⊗ F e B † rig [1 /t ]) ⊗ B e,E W e = ( E ⊗ F e B † rig [1 /t ]) ⊗ E ⊗ F B † rig ,K D so that, taking the invariants by ϕ q , we get that W e ≃ W ( D ) as B e,E -representations.Let τ ∈ Σ. By definition of W + dR,τ ( D ), we have W + dR,τ ( D ) = ( E ⊗ F B +dR ) ⊗ ι g D r n ( g ) forsome g ∈ Ω r,τ with r big enough, and hence W + dR,τ ( D ) = ( E ⊗ F B +dR ) ⊗ ι g f D r n ( g ) where f D r = f D r ( W ) = ( E ⊗ F e B † ,r rig ) ⊗ E ⊗ F B † ,r rig ,K D r by proposition 5.5. Recall that f D r ( W ) = n y ∈ ( E ⊗ F e B † ,r rig [1 /t ]) ⊗ B e,E W e such that ι g ( y ) ∈ W + dR,τ ( g ) | F for all g ∈ Ω r o , so that, after tensoring by E ⊗ F B +dR over ι g , we get W + dR,τ ( D ( W )) = W + dR,τ .Let D be an F -analytic ( ϕ q , Γ K )-module and let W = W ( D ) and f D = ( E ⊗ F B † rig ) ⊗ E ⊗ F B † rig ,K D . The same reasoning as above shows that( E ⊗ F e B † rig [1 /t ]) ⊗ E ⊗ F B † rig ,K D = ( E ⊗ F e B † rig [1 /t ]) ⊗ E ⊗ F B † rig ,K D ( W ( D ))and that( E ⊗ F e B † rig [1 /t ]) ⊗ E ⊗ F B † rig ,K f D = ( E ⊗ F e B † rig [1 /t ]) ⊗ E ⊗ F B † rig ,K f D ( W ( D )) . If M is a ( ϕ q , Γ K )-module over E ⊗ F B † rig , note that we can recover M inside M [1 /t ]by M = (cid:26) x ∈ M [1 /t ] such that ι g ( x ) ∈ ( E ⊗ F B +dR ) ⊗ ι g E ⊗ F e B † rig M for all g with n ( g ) ≫ (cid:27) . In particular, since( E ⊗ F e B † rig [1 /t ]) ⊗ E ⊗ F B † rig ,K f D = ( E ⊗ F e B † rig [1 /t ]) ⊗ E ⊗ F B † rig ,K f D ( W ( D )) , this shows that f D = f D ( W ( D )) . NOTE ON F -ANALYTIC B -PAIRS Since D is F -analytic, we have ∇ τ (( f D K ) pa ) ⊂ t τ · ( f D K ) pa ) for all τ ∈ Σ \{ id } by proposition4.3, hence there exists, still by proposition 4.3, a unique F -analytic ( ϕ q , Γ K )-module D † rig over E ⊗ F B † rig ,K such thatSol( f D pa K ) = ( E ⊗ F ( e B † rig ,K ) F -pa ) ⊗ E ⊗ F e B † rig ,K D † rig and such that f D = ( E ⊗ F e B † rig ,K ) ⊗ E ⊗ F B † rig ,K D † rig In particular, we have D = D ( W ( D )) = D † rig , which concludes the proof.
6. A simpler equivalence in the F -analytic case In this section, we quickly recall the constructions of Ding’s B σ -pairs and explain thecorresponding equivalence of category that follows between F -analytic ( ϕ q , Γ K )-modulesand B id -pairs.Let B LT e,F = ( e B +rig [1 /t π ]) ϕ q =1 . As usual, it is easy to check that B LT e,F = ( e B † rig [1 /t π ]) ϕ q =1 .Following [ Din14 ], we make the following definition:
Definition 6.1 . — 1. Let σ ∈ Σ E be any embedding. A B σ -pair is the data of acouple W σ = ( W LT σ,E , W +dR ,σ ) where W LT σ,E is a finite free E ⊗ σF B LT e,F -module equippedwith a semi-linear G K action and W +dR ,σ is a G K -invariant B +dR ,σ -lattice in W dR ,σ := W LT σ,E ⊗ E ⊗ σF B LT e,F B dR ,σ .2. For two B σ -pairs W σ , W ′ σ , a morphism f : W σ −→ W ′ σ is a G K -invariant E ⊗ σF B LT e,F -linear map f LT σ,E : W LT σ,E −→ ( W ′ σ,E ) LT such that the induced B dR ,σ -linear map f dR ,σ := f LT σ,E ⊗ id : W dR ,σ −→ W ′ dR ,σ sends W +dR ,σ to ( W ′ ) +dR ,σ .Let W = ( W e , W +dR ) be a ( B, E )-pair. Let W LT σ,E = n w ∈ W e : τ ( w ) ∈ W +dR ,σ ◦ τ − for all τ ∈ Gal( E/ Q p ) , τ | F = id o . By [
Din14 , Lemm. 1.3], this is a E ⊗ σF B LT e,F -module. Proposition 6.2 . —
For σ ∈ Σ E , the functor F σ : { ( B, E ) − pairs }−→{ B σ − pairs } given by W = ( W e , W +dR ) W σ = ( W LT σ,E , W +dR ,σ ) induces an equivalence of categories.Proof . — This is [ Din14 , Prop. 3.7]. LÉO POYETON
For σ ∈ Σ E , let G σ denote the inverse functor of F σ defined by Ding in [ Din14 , Lemm.3.8]. We say that a B id -pair W is F -analytic if for all σ ∈ Σ E such that σ | F = id F ,then W +dR ,σ /tW +dR ,σ is the trivial C p -representation of G K , where W +dR ,σ is the secondcomponent of the B σ -pair F σ ◦ G id ( W ). By [ Din14 , Lemm. 3.9], this is the same asasking that the corresponding (
B, E )-pair G id ( W ) is F -analytic. Proposition 6.3 . — If D is a ( ϕ q , Γ K ) -module over E ⊗ F B † rig ,K , free of rank d , then1. W LTid ,E ( D ) = ( E ⊗ F e B † rig ,K [1 /t π ] ⊗ B † rig ,K D ) ϕ q =1 is a free E ⊗ σF B LT e,F -module of rank d which is G K -stable;2. W +dR , id = ( E ⊗ F B +dR ) ⊗ ι g E ⊗ F B † ,rn ( g )rig ,K D r n ( g ) ! g ∈ Ω id ,r does not depend on n ( g ) ≫ andis a free B +dR , id -module of rank d which is G K -stable.3. W ( D ) LT = ( W LTid ,E ( D ) , W +dR , id ( D )) is a B id -pair. Moreover, if D is F -analytic, thenso is W ( D ) .Proof . — The proof of items 1, 2 and 3 is the same as in 5.9. The part on F -analyticitynow follows from the remark above and the fact (which follows easily from proposition6.2) that the B id -pair W ( D ) we just constructed is exactly F id ( W ′ ) where W ′ is the( B, E )-pair attached to D constructed in proposition 5.9.In particular, this construction is the exact analogue of Berger’s construction [ Ber08 ,Prop. 2.2.6] in the cyclotomic case.We now explain how to recover the ( ϕ q , Γ K )-module D † rig attached to an F -analytic B id -pair W . Let W id = ( W LTid ,E , W +dR , id ) be an F -analytic B id -pair. Lemma 6.4 . —
Let f D r ( W ) LT = n y ∈ ( E ⊗ F e B † ,r rig [1 /t π ]) ⊗ E ⊗ F B LT e,F W LTid ,E such that ι g ( y ) ∈ W +dR , id for all g ∈ Ω r with τ ( g ) = id o . Then:1. f D r ( W ) LT is a free E ⊗ F e B † ,r rig -module of rank d ;2. f D r ( W ) LT [1 /t π ] = E ⊗ F e B † ,r rig [1 /t π ] ⊗ E ⊗ F B LT e,F W LTid ,E .Proof . — This is the same proof as in lemma 5.4 but here we do not need to keep trackof all the embeddings.We know that there are enough pro-analytic vectors inside f D ( W ) LT , just because wealready know by the constructions of §5 that it contains the F -analytic ( ϕ q , Γ K )-module D ( W ′ ) attached to W ′ = G id ( W ) of theorem 5.7. We can now recover it by taking the NOTE ON F -ANALYTIC B -PAIRS pro-analytic vectors of f D ( W ) LT and taking the module D † rig ( f D ( W ) LT ) given by propo-sition 4.3. In particular, the following is a straightforward consequence of our previousconstructions: Theorem 6.5 . —
The functors D W ( D ) LT and W id D † rig ( f D ( W ) LT ) are inverse ofeach other an give rise to an equivalence of categories between the category of F -analytic ( ϕ q , Γ K ) -modules and the category of F -analytic B id -pairs. References [BC16] Laurent Berger and Pierre Colmez,
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November 18, 2020
Léo Poyeton , BICMR, Peking University • E-mail : [email protected]