Local constancy for reductions of two-dimensional crystalline representations

Local constancy for reductions of two-dimensional crystalline representations

Emiliano Torti August 24, 2020

Abstract

We prove the existence of local constancy phenomena for reductions in a general (odd) prime power setting of two-dimensional irreducible crystalline representations of Gal(Qp/Qp). These rep- resentations depend on two parameters: a traceapand a weightk. They appear for example in the context of classical modular forms of tame level. We find an (explicit) local constancy result with respect toapusing Fontaine’s theory of (ϕ,Γ)-modules; its crystalline refinement due to Berger via Wach modules and their continuity properties. The local constancy result with respect to k (for

ap 6= 0) will follow from a local study of Colmez’s rigid analytic space parametrizing trianguline

representations. This work extends some results of Berger obtained in the residually semi-simple case.

1 Introduction

Crystalline representations play a central role in the study ofp-adic representations of the local absolute Galois groupG_Qp:= Gal(Qp/Q^p) (see, for example, some density results due to Berger (see Thm. IV.2.1 in [Ber04]), Chenevier (see Thm. A in [Che13]), Colmez (see sec. 5.1 in [Col08]), and Kisin (see Thm.

0.3 in [Kis10])). We are interested in studying irreducible crystalline representations ofG_Qpof dimension two and in particular their reductions modulo prime powers.

Letpbe an odd prime, let k ≥2 be an integer and ap ∈ m_E where Eis a finite extension of Qp, m_E denotes the maximal ideal of the ring of integersO_Ewith residue fieldk_E. Fix once and for all a choice of a uniformizer, sayπ_E. LetDk,a_p:=Ee1⊕Ee2 be the filteredϕ-module whose structure is given by:

ϕ=

0 −1 p^k−1 ap

and a filtration Filⁱ(Dk,a_p) =







Dk,a_p ifi≤0

Ee1 if 1≤i≤k−1 0 ifi≥k

By a theorem of Colmez and Fontaine (see Thm. A in [CF00]), there exists a unique crystalline irreducibleE-linear representationVk,a_pof dimension two, with Hodge-Tate weights{0, k−1}such that Dcris(V_k,a^∗

p) =Dk,a_p, whereV_k,a^∗

pdenotes theE-linear dual representation ofVk,a_p. By a result of Breuil (see Prop. 3.1.1 in [Bre03]), up to twist, any irreducible two-dimensional crystalline representation is isomorphic toVk,a_p for somek≥2 andap∈m_E.

These results give rise to the natural questions of whether it is possible to completely classifyVk,a_p in terms ofkand ap, and how Vk,ap varies when the parameterskandap vary p-adically.

In general, classifying the representationsV_k,ap in characteristic zero turns out to be an hard problem even though some progress have been made in particular cases via the local Langlands correspondence (e.g. see [Pas09]). Nevertheless, much progress has been made in describing the semi-simple residual reductions of the representationsV_k,apusing different approaches. We will briefly recall the state of art in the residual case. Consider the E-linear representation V_k,ap and let T_k,ap be a G_Qp-stable lattice inside Vk,a_p, we have an isomorphism Tk,a_p ⊗_OE E =∼ Vk,a_p of G_Qp-modules. Denote by Vk,a_p the

semi-simplification ofTk,a_p⊗_OEk_E; by the Brauer-Nesbitt’s theorem, the representation Vk,a_p does not depend on the chosenG_Qp-stable latticeTk,a_p.

The problem of describing the representations Vk,a_p has been deeply studied by many authors via thep-adic and mod pLanglands correspondence (see for example [Bre03] and [BG09]), via Fontaine’s theory of (ϕ,Γ)-modules and its crystalline refinement via Wach modules (see for example [BLZ04]) or via deformation theory (see for example [Roz17])). However, the problem of classifying them is still open and only partial results are known (see for example [BLZ04], [BG15], [BGR18]) although partial conjectures have been formulated (see Conj. 1.5 in [Bre03] and Conj. 1.1 in [Gha19]).

In order to try to describe the reductionsVk,ap, one different approach consists in finding isomorphisms between different residual representations of the formV_k,ap when we letkanda_p varyp-adically. This approach has been developed by Berger et al. with the so-called local constancy results both in the trace and in the weight (see Thm. A and Thm. B in [Ber12] and for for the casea_p= 0 see Thm. 1.1.1 in [BLZ04]).

The purpose of this article is to extend Berger’s result to a prime power setting. The main difficulty lies in keeping track of the Galois stable lattices involved in the congruences because no semi-simplification process is, a priori, allowed (or defined) for general reductions modulo prime powers. Hence, in proving the existence of such congruences, a dependency on a choice of the Galois stable lattices is expected;

but as we will see later, in some cases, the result will be independent of such choice.

The first result of the article is the following local constancy result with respect to the trace, i.e. we fix the weight and we let the trace of the crystalline Frobenius varyp-adically:

Theorem 1.1. (Local constancy with respect toap)

Letap, a⁰_p∈m_Eandk≥2be an integer. Letm∈ ¹_e(Z≥1)such thatv(ap−a⁰_p)≥2·v(ap) +α(k−1) +m, then for every G_Qp-stable lattice Tk,ap inside Vk,ap there exists a G_Qp-stable lattice Tk,a⁰_p inside Vk,a⁰_p

such that

T_k,ap⊗O_EO_E/(p^m)∼=T_k,a⁰

p⊗O_EO_E/(p^m)asG_Qp-modules;

whereα(k−1) =P

n≥1b_pn−1^k−1(p−1)c.

This result will be proven in two steps. First we will prove that ifapanda⁰_pare sufficientlyp-adically close then it is possible to deform p-adically the Wach module attached to the representationTk,a_p into a new Wach module which will correspond to the representationTk,a⁰_p. Afterwards, we will prove that if the two Wach modules involved arep-adically close (in some sense that will be clarified later) then the corresponding representationsTk,a_p and Tk,a⁰_p will bep-adically close as well. We will refer to this feature as continuity property of the Wach modules.

The second result of the article is the following local constancy result with respect to the weight, i.e.

we fix the trace of the crystalline Frobenius and we let the weight vary in a neighborhood of the weight spaceW:

Theorem 1.2. (Local constancy with respect tok)

Leta_p∈m_E− {0} for some finite extensionEof Qp. Letk≥2 andm∈ ¹_e(Z≥1)be fixed. Assume that k≥(3v(a_p) +m)·

1− p

(p−1)² −1

+ 1.

There exists an integerr=r(k, ap)≥1 such that ifk⁰−k∈p^r+m(p−1)Z≥0 then there existG_Qp-stable latticesT_k,ap ⊂V_k,ap andT_k0,ap ⊂V_k0,ap such that

T_k,ap⊗O_EO_E/(p^m)∼=T_k⁰_,ap⊗O_EO_E/(p^m)asG_Qp-modules.

The idea is to prove that the representations V_k,ap and V_k⁰_,ap are respectively congruent modulo p^m to two representationsV

k,a_p+^pk−1

ap

and V

k⁰,ap+^pk_ap^{0 −1} which fit into an analytic family of trianguline representations in the sense of Berger and Colmez (see [BC08]); as a consequence, the claims will follow

from proving that if k and k⁰ are sufficiently close in the weight space W then the representations Vk,a_p+^pk−1

ap

andV

k⁰,a_p+^{pk0 −1}

ap

arep-adically close as well (in a sense that will be clarified precisely later in the article). The result constitutes a converse (in a particular crystalline case) to a non-published theorem of Wintenberger, also proven by Berger and Colmez (see Thm 7.1.1 and Cor. 7.1.2 in [BC08]), concerning the continuity property of the Sen periods and the Hodge-Tate weights.

Specializing the above theorems to the casem= 1/e, we get a slightly stronger result than the known local constancy results in the semi-simple residual case (see Thm. A and Thm. B in [Ber12]); indeed our conclusions do not involve any semi-simplification process, so for example, being residually decom- posable (i.e. direct sum of two characters modulop) for some choice of lattice is also a locally constant phenomenon.

The motivation behind the study of local constancy phenomena modulo prime powers is two-fold. From a purely representation theoretical point of view, the interest in understanding reductions modulo prime powers of crystalline representations lies in the result of Berger on limits of crystalline representations (see [Ber04]). To be more precise, Berger’s result implies that ifV is any p-adic representation ofG_Qp with Hodge-Tate weights in a bounded interval I and if {Vi}_i∈I is a countable family of crystalline representations with HT weights inI such thatT ≡Ti modpⁱ, whereT is a fixedG_Qp-stable lattice inV andTi is aG_Qp-stable lattice inVi, thenV is also crystalline.

Moreover, we observe that a good source of examples for the crystalline representations of the form V_k,ap comes from restriction at G_Qp of Galois representations attached to classical modular forms of tame level. To be precise, letf be a classical normalized cuspidal eigenform inS_k(Γ₀(N)) whereN is a positive integer prime withpand denote by ρ_f its attachedp-adic Galois representation constructed by Deligne and Shimura. We define by V_p(f) := ρ_f

_G

Qp

the restriction of ρ_f at the decomposition group atp. It is well-known (see [Sch90]), that under the mild hypothesis a²_p 6= 4p^k−1 (see [CE98]), theG_Qp-representation Vp(f) is crystalline and moreover we have that Dcris(Vp(f)^∗) = Dk,ap and so V_p(f)∼=V_k,ap where a_p is thep-th coefficient of theq-expansion off. A straightforward application of the results in this article consists in using the explicit local constancy results in the trace (see Thm. 1.1 and Cor. 4.6) to find upper and lower bounds for the number of non-isomorphic classes of reductions modulo prime powers of modular crystalline representations ofG_Qpcoming from classical modular forms of tame level.

The paper is organized as follows. In Section 2, we will recall the notions of (ϕ,Γ)-module of Fontaine and of Wach module and their main properties which will be used later in the article. In Section 3, we will recall the continuity property of Wach modules which will play a key role in proving the local constancy result in the trace. In Section 4, we will show how top-adically deform Wach modules and we will state and prove the explicit local constancy result when the weight k is fixed and we let the trace of the crystalline Frobeniusapvary. Finally, in Section 5, we are going to state and prove the local constancy result when we fix the trace of the crystalline Frobeniusap and we let the weightkvary.

Acknowledgments: This work has been developed during a PhD at the University of Luxembourg.

I would like to thank my advisor G. Wiese for the help during the writing of this article. Special thanks go to L. Berger for allowing me to visit him at the ENS in Lyon and for many enlightening conversations.

I would also like to thank A. Conti, L. Demb´el´e, A. Maksoud and A. Vanhaecke for many interesting remarks.

2 Wach modules and crystalline representations

Letpbe an odd prime and letE⊆Qpbe a finite extension ofQp. We denote byO_Ethe ring of integers of E, by π_E a uniformizer, by k_E the residue field and denote by e the ramification index of E over Qp. Let Γ be a group isomorphic to Z^×p via a map χ : Γ → Z^×p. Fix once and for all a topological generator of Γ (which is procyclic asp6= 2), sayγ. For the sake of completeness, we will briefly recall the construction of some of the rings of Fontaine necessary for introducing the (ϕ,Γ)-modules and the

theorem characterizing their relation with certain Galois representations ofG_Qp. Let{⁽ⁿ⁾}_n≥1⊂ ¯

Q^p be a system of roots of unity such that:

(1) ⁽¹⁾ 6= 1,

(2) ⁽ⁿ⁾∈µ_pn ⊂Qp, (3) (⁽ⁿ⁺¹⁾)^p=⁽ⁿ⁾.

One can think of:= (⁽¹⁾, ⁽²⁾, . . .) as an element of Fontaine’s ringE = lim

←−Cp (with projective limit maps given by the Frobenius mapsz7→z^p) where Cp is the p-adic completion ofQp. It is well known thatE is an algebraically closed field of characteristicp. Now, consider the fieldsQ^(n)p :=Q^p(⁽ⁿ⁾) and defineQ^(∞)p =∪n≥1Q^(n)p . Denote byH_Qp the Galois group Gal(Qp/Q^(∞)p ).

Thep-adic cyclotomic characterω gives the exact sequence:

1−→H_Qp−→G_Qp−→^ω Γ∼=Z^×p −→1.

Consider the field F = Fp((−1)) inside E. Let A_Qp be the p-adic completion of Zp[[x]][¹_x]; it is a complete discrete valuation ring whose residue field can be identified withF(one can identifyxwith a suitable Teichmuller lift of−1). LetAbe thep-adic completion of the strict henselizationA^sh_Q

pofA_Qp inside ˜A:=W(E). Note thatA^sh_Qp can be identified with the ring of integers of the maximal unramified extension of the fieldA_Qp[¹_p] inside ˜A[¹_p].

The Galois group G_Qp acts on E by acting on Cp and by functoriality on the projective limit. By functoriality of the Witt vectors, the groupG_Qp also acts on ˜A=W(E) and we have that Ais G_Qp- stable. It is also true thatA^HQp =A_Qp.

Now, we defineA_E as A_Qp⊗_ZpO_E, by a result of Dee (see prop. 2.2.2 in [Dee01]) we have that A_E is isomorphic to theπ_E-adic completion of O_E[[x]][_x¹]. One can think ofA_E inside the ring A^(OE) :=

A ⊗_ZpO_E. The ringA_E inherits aG_Qp-action via the natural action ofG_Qp onAand trivial action on O_E. It follows that (A^(OE))^HQp =A_Eand so A_Ehas a structure of Γ-module.

Hence, the ring A_E has a natural O_E-linear action of Γ and a O_E-linear Frobenius endomorphism ϕ given by the following expressions:

ϕ(f(x)) =f((1 +x)^p−1) for allf(x)∈ A_E,

η(f(x)) =f((1 +x)^χ(η)−1) for allf(x)∈ A_E, for allη∈Γ.

Finally, we can recall the following:

Definition 2.1. An ´etale (ϕ,Γ)-module D over O_E is an A_E-module of finite type endowed with a semilinear Froebenius mapϕsuch thatϕ(D)generatesD asA_E-module (this is the ´etale property) and a semilinear continuous action ofΓwhich commutes withϕ.

The category of ´etale(ϕ,Γ)-module overO_E will be denoted by Mod^´et_(ϕ,Γ)(O_E).

Denote byRep_O

E(G_Qp) the category of O_E-linear representations of G_Qp, i.e. the category ofO_E- modules of finite type with a continuousO_E-linear action ofG_Qp.

By a theorem of Fontaine (see A.3.4 in [Fon90]) and its generalization by Dee (see 2.2 in [Dee01]) we have the following:

Theorem 2.2. There exists a natural isomorphism D:Rep_O

E(G_Qp)→Mod^´et_(ϕ,Γ)(O_E) given byD(T) = (A^(OE)⊗_OET)^HQp, whereA^(OE):=A ⊗_ZpO_E.

A quasi-inverse functor, which is a natural isomorphism as well, is given by:

T:Mod^´et_(ϕ,Γ)(O_E)→Rep_O

E(G_Qp) given byT(D) = (A^(OE)⊗_AED)^ϕ=1.

Remark 1. Note that the equivalence of categories given by the above theorem preserves the objects killed by a fixed power of a chosen uniformizer. This essentially follows from the exactness of the functor D(see prop. 2.1.9 in [Dee01]) and so (πⁿ_E)·D(T) =D((πⁿ_E)·T) in the categoryMod^´et_(ϕ,Γ)(O_E). Same goes for the quasi-inverse functorT(see prop. 2.1.24 in [Dee01]).

Remark 2. LetMod^´et,tors_(ϕ,Γ) (O_E) andMod^´et,free_(ϕ,Γ)(O_E) be respectively the categories of torsion and free (asA_E-modules) ´etale (ϕ,Γ)-modules. There is a notion of Tate dual for such ´etale (ϕ,Γ)-modules. Let B_E beA_E[¹_p] and define the Tate dual as follows (see sec. I.2 in [Col10]):

ifD∈Mod^´et,tors_(ϕ,Γ) (O_E) thenD^∗:= Hom_AE

D,B_E/A_E dx 1 +x

ifD∈Mod^´et,free_(ϕ,Γ)(O_E) thenD^∗:= Hom_AE

D,A_E dx 1 +x

whereB_E/A_E1+x^dx is the inductive limit ofp⁻ⁿA_E/A_E1+x^dx in the categoryMod^´et,tors_(ϕ,Γ) (O_E) and the struc- ture of (ϕ,Γ)-module is given by:

γ dx 1 +x

=χ(γ) dx

1 +x andϕ dx 1 +x

= dx 1 +x.

Note the important fact that these two notions of dual are compatible in the following sense: if D ∈ Mod^´et,free_(ϕ,Γ)(O_E) then for alln∈ ¹_e(Z≥1) we have thatD^∗/pⁿD^∗∼= (D/pⁿD)^∗ (see prop I.2.5 in [Col10]).

The ´etale (ϕ,Γ)-modules corresponding to crystalline representations can be described more explic- itly via the theory of Wach modules developed by Berger and Wach. We will briefly recall the definition of Wach modules and their main properties. First, we defineA⁺_E =O_E[[x]] inside A_E. It inherits nat- urally the actions ofϕand Γ by restriction fromA_E. Note also thatA_E is obtained by takingπ_E-adic completion of the localizationO_E[[x]][1/x] ofA⁺_E =O_E[[x]] at the multiplicative set{1, x, x², . . .}. More- over, sinceA_E is Noetherian, we have thatA_E is flat asA⁺_E-module since localization and completion preserve such property. Following Berger (see [Ber12]), we have the following

Definition 2.3. A Wach module of height h ≥ 1 is a free A⁺_E-module of finite rank endowed with commutativeA⁺_E-semilinear actions of a Frobenius mapϕ and of the groupΓ such that:

(1) D(N) :=A_E⊗_A+ E

N ∈ Mod^´et_(ϕ,Γ)(O_E), (2) Γacts trivially onN/xN,

(3) N/ϕ^∗(N)is killed by Q^h,

whereϕ^∗(N)denotes the A⁺_E-module generated byϕ(N), andQ= ^(1+x)_x^p−1 ∈ A⁺_E.

We recall that the Wach modules are the right linear algebra objects to specialize Fontaine’s equiv- alence to crystalline representations. Indeed, we have the following (see Prop. 1.1 in [Ber12]):

Proposition 2.4. LetN be a Wach module of heighth. TheE-linear representationE⊗_OET(A_E⊗_A+ E

N) ofG_Qp is crystalline with Hodge-Tate weights in the interval [−h; 0]; and

Dcris(E⊗O_ET(A_E⊗_A+ E

N))∼=E⊗O_EN/xN asϕ-modules.

Moreover, all crystalline representations with Hodge-Tate weights in[−h; 0] arise in this way.

3 Continuity of the Wach modules

Berger proved (see sec. III.4 or Thm. 2 in [Ber04]) that the there exists an equivalence of categories between rational Wach modules (overE⊗_OEA⁺

E) andG_Qpcrystalline representations. As a consequence,

Berger proved that if we fix aE-linear representationV ofG_Qpand denote byDits corresponding ´etale (ϕ,Γ)-module via Fontaine’s functor there is an inclusion preserving bijection between lattices insideV and Wach modules (over A⁺

E) contained in D and which are A⁺

E-lattices. Denote byN such bijective map that associates to eachG_Qp-stable lattice of a crystalline representation its corresponding Wach module.

In this section, we will prove that in some natural sensep-adically close Wach modules will correspond (viaN) to p-adically closeO_E-linear representations sitting in crystalline representation and viceversa.

We will start by clarifying what we mean byp-adically close Wach modules. Given two Wach modulesN1

andN2we say thatN1andN2are congruent modulo some prime power, i.e. N1≡N2 modπ_E^mfor some m∈Z≥1, if there exists a A⁺_E-module isomorphism betweenN1⊗_A+

E

A⁺_E/(π^m_E) andN2⊗_A+ E

A⁺_E/(π_E^m) which is (ϕ,Γ)-equivariant.

Note that, essentially by definition, we have thatN₁≡N₂ modπ^m

E if and only if there exist basis of N₁andN₂asA⁺_E-modules such that, after definingP₁= Mat(ϕ|N1),P₂= Mat(ϕ|N2),G₁= Mat(γ|N1), G₂ = Mat(γ|_N2) (note thatP₁, P₂, G₁, G₂ ∈M_d×d(A⁺

E) whered= rank_A+ E

(N_i) for i= 1,2) it follows that:

(P1≡P2 modπ^m_E, G₁≡G₂ modπ^m

E.

We have the following continuity result (this is Thm. IV.1.1 in [Ber04]):

Proposition 3.1. LetT₁andT₂be two Galois stable lattices inside respectively two crystallineE-linear representation V₁ and V₂ of Hodge-Tate weights inside [−r; 0], and assume there is an n ∈ ¹_e(Z≥1) with n ≥ α(r) such that T1⊗_OE O_E/(pⁿ) ∼= T2⊗_OE O_E/(pⁿ) as G_Qp-modules, then N(T1) ≡ N(T2) modp^n−α(r).

Now, if N is a Wach module denote by T(N) := T(D(N)) the O_E-linear representation of G_Qp attached toN. We recall thatT(N)⊗_OEEis a crystalline representation.

We are interested in the following result:

Proposition 3.2. LetN1 andN2be two Wach modules (overO_E) with the same rank asA⁺_E-modules.

Assume thatN1≡N2 modπⁿ_E for somen∈Z≥1. ThenT(N1)⊗O_EO_E/(π_Eⁿ)∼=T(N2)⊗O_EO_E/(π_Eⁿ)as G_Qp-modules.

Proof. Consider the ´etale (ϕ,Γ)-modules D(Ni) = Ni⊗_A+ E

A_E for i = 1,2. Since A_E is flat as A⁺

E- module (module structure given by inclusion) we have the following chain of isomorphisms of torsion

´etale (ϕ,Γ)-modules:

D(N1)/πⁿ_ED(N1)∼=N1/π_EⁿN1⊗_A+ E

A_E∼=N2/πⁿ_EN2⊗_A+ E

A_E∼=D(N2)/πⁿ_ED(N2).

Now, the claim follows just by applying Fontaine’s functorT, which is exact (see Theorem 2.2 and see Remark 1).

4 Local constancy with respect to the trace

In this section we are going to prove an explicit local constancy result with respect to the trace (i.e. the weightkwill be fixed) for reductions modulo prime powers of representations of the typeVk,a_p.

4.1 Some linear algebra of Wach modules

As explained in the previous section, a congruence between Wach modules (modulo some prime power) can be translated into a congruence (modulo the same prime power) between systems of matrices

representing the (ϕ,Γ) actions on the Wach modules involved. In this section, we will see how to p- adically deform a Wach module into another one via linear algebra means for the systems of matrices associated to the (ϕ,Γ)-module structure.

As long as it will be possible, we will keep the same notation as Berger (see [Ber12]). We recall thatp is an odd prime and Eis a finite extension of Qp with ramification indexe. Let v be the normalized p-adic valuation (i.e. v(p) = 1). Letr≥1 be a integer and defineα(r) :=Pr

j=1v(1−χ(γ)^j) where we recall thatγ is a fixed topological generator of the pro-cyclic group Γ. The constant α(r) has also an explicit description given byα(r) =P

n≥1b_pn−1^r(p−1)cfor a proper choice ofγ (see [Ber12]).

We start by recalling two useful results in linear algebra (see Lemma 2.1 in [Ber12]):

Lemma 4.1. If P0 ∈M2(O_E) is a matrix with eigenvaluesλ6=µ, and if δ=λ−µ, then there exists Y ∈M2(O_E) such thatY⁻¹∈δ⁻¹M2(O_E)and Y⁻¹P0Y =

λ 0 0 µ

. and the following corollary (see cor. 2.2 in [Ber12]):

Corollary 4.2. If α ∈ ¹_eZ≥0 and ∈ O_E are such that v() ≥ 2v(δ) +α, then there exists H₀ ∈ p^αM2(O_E)such that det(Id+H0) = 1 and Tr(H0P0) =.

The corollary above represents the starting point to deform a Wach module into another one. Given the matrixP0, it gives ap-adically small matrixH0 such that the productH0P0will have a prescribed p-adically small trace. In practice, this will be applied when P0 is obtained by the action of ϕ on Dcris(Vk,a_p) for somek≥2 andap∈m_E.

The idea behind the next results is to show howH0 gives rise to a deformation of a whole system of matrices (attached to a Wach module) to a p-adically close one preserving the characterizing linear algebra properties of the action onϕand Γ on the Wach module.

We have the following result (it is a little generalization of Prop. 2.3 in [Ber12]):

Proposition 4.3. Let m ∈ ¹_eZ≥0 and d ∈ Z≥2. If G ∈ Id+xMd(O_E[[x]]) and k ≥ 2 and H0 ∈ p^α(k−1)+mMd(O_E), then there existsH ∈p^mMd(O_E[[x]])such thatH(0) =H0andHG≡Gγ(H)modx^k. Proof. AsG∈Id +xMd(O_E[[x]]), we can writeG= Id +xG1+x²G2+. . . where Gi∈Md(O_E) for all i∈ Z≥1. We prove that for any positive integer r, there exists anHr ∈pα(k−1)−α(r)+mMd(O_E) such that if we defineH =H0+xH1+x²H2+· · ·+x^k−1Hk−1we have thatHG≡Gγ(H) mod x^k. We start fromr= 1, then sinceγ(H) =H0+γ(x)H1+γ(x²)H2+. . . γ(x^k−1)Hk−1and for allw∈Z≥1we haveγ(x^w) = ((1+x)^χ(γ)−1)^w, we deduce that we can defineH1such that (1−χ(γ))H1=G1H0−H0G1. Since by hypothesis H₀ ≡ 0 modp^α(k−1)+m and by definition α(1) = v_p(1−χ(γ)), we deduce that H₁∈pα(k−1)−α(1)+mM_d(O_E).

Using now the same argument, one can actually see howH_ris uniquely determined byH₀, H₁, . . . H_r−1 and moreover (1−χ(γ)^r)H_r∈pα(k−1)−α(r−1)+mM_d(O_E), note thatα(r) =α(k−1)−α(r−1). To be precise, we prove this by induction onr≤k−1. The first caser= 1 is proven above, now assume the caser−1, we are going to prove the statement forr.

It is straightforward to prove that, expanding the expression HG ≡ Gγ(H) modx^k, the following identity holds:

(1−χ(γ)^r)Hr=

r−1

X

h=0

X^h

n=0

γ(xⁿ)hHn

G_r−h−

r−1

X

i=0

HiG_r−i, forr≤k−1

whereγ(xⁿ)h is theh-th coefficient (i.e. coefficient ofx^h) of the polynomialγ(xⁿ). Note now that the mapα(n) is non-decreasing asngrows. Hence, by inductive hypothesis, we deduce that for anyisuch that 0≤i≤r−1 we haveHi≡0 mod (pα(k−1)−α(r−1)+m). Sincev(1−χ(γ)^r) +α(r−1) =α(r), we deduce thatHr≡0 mod (pα(k)−α(r)+m). This concludes the proof.

The following result completes the linear algebra deformation process of a system of matrices that will represent a (ϕ,Γ)-action on Wach modules:

Proposition 4.4. Letm∈ ¹_eZ≥0andd∈Z≥2. LetG∈Id+xMd(O_E[[x]])andP ∈Md(O_E[[x]])satisfy P ϕ(G) =Gγ(P)and det(P) =Q^k−1 whereQ= ^(1+x)_x^p−1.

IfH0∈p^α(k−1)+mMd(O_E), then there existG⁰∈Id+xMd(O_E[[x]])andH ∈p^mMd(O_E[[x]])such that:

(1) H(0) =H0;

(2) P⁰ϕ(G⁰) =G⁰γ(P⁰), where P⁰ = (Id+H)P; (3) P ≡P⁰ modp^m;

(4) G≡G⁰ modp^m.

Proof. After applying the previous proposition for the existence of the matrixH which satisfiesH(0) = H0, the existence ofG⁰ follows directly from Prop. 2.4 in [Ber12]. Hence the claims (1) and (2) hold.

The claim (3) is clear sinceH ∈p^mMd(O_E[[x]]) implies thatP ≡P⁰ modp^m. What it is left to prove is (4), i.e. G≡G⁰ modp^m. In order to prove this, we need to look at how the matrixG⁰ is defined.

The matrix G⁰ is constructed as an x-adic limit inside Id +xMd(O_E[[x]]) (note that we are dealing with non-commutative rings). Define G⁰_k := G and observe that it satisfies by construction G⁰_k − P⁰ϕ(G⁰_k)γ(P⁰)⁻¹ = x^kRk for some Rk ∈ Md(O_E[[x]]). Then define G⁰ as the x-adic limit of G⁰_j, for j ≥ k, which satisfies G⁰_j+1 =G⁰_j+x^jS_j for some S_j ∈M_d(O_E), and G⁰_j−P⁰ϕ(G_j⁰)γ(P⁰)⁻¹ =x^jR_j whereR_j ∈M_d(O_E[[x]]).

We will prove by induction that:

(i) R_j ≡0 modp^m, (ii) G⁰_j≡Gmodp^m.

First the casej =k: sinceH ≡0 modp^m then P ≡P⁰ modp^m which implies thatRk ≡0 modp^m, and by construction G⁰_k =G so the first case of induction is done. Now assume j ≥ k and that the above claims hold forj, we will prove them forj+ 1.

We have that there existsSj∈Md(O_E) such that:

G⁰_j+1−P⁰ϕ(G⁰_j+1)γ(P⁰)⁻¹=G⁰_j+x^jSj−P⁰ϕ(G⁰_j)γ(P⁰)⁻¹−P⁰x^jQ^jSjγ(P⁰)⁻¹=

=x^j(Rj+Sj−Q^j−k+1P⁰SjQ^k−1)γ(P⁰)⁻¹)∈x^j+1Md(O_E[[x]]).

Note that we used thatϕacts trivially on Md(O_E) and thatϕ(x^j) =x^jQ^j for allj∈Z≥1. We want to prove that there existsSj∈p^mMd(O_E) such that:

Rj+Sj−Q^j−k+1P⁰SjQ^k−1γ(P⁰)⁻¹∈xMd(O_E).

Evaluating the above expression at x = 0, the claim is equivalent to prove that there exists Sj ∈ p^mMd(O_E) such that:

Sj−p^j−k+1P⁰(0)Sjp^k−1(γ(P⁰)⁻¹)(0) =−Rj(0).

Now, since R_j ≡ 0 modp^m, we have that R_j(0) ≡ 0 modp^m. It is clear that the map S 7→ S− p^j−k+1P⁰(0)Sp^k−1(γ(P⁰)⁻¹)(0) gives a bijection of M_d(O_E). Moreover, it is also clear that it is a bijection on p^mM_d(O_E). As R_j(0) ≡ 0 modp^m, we have the existence of S_j ≡ 0 modp^m such that the above relations are satisfied. By inductive hypothesisR_j≡0 modp^m, soR_j+1≡0 modp^m. Since S_j≡0 modp^mimplies thatG⁰_j+1=G⁰_j+x^jS_j ≡Gmodp^m. This concludes the proof.

4.2 Local constancy modulo prime powers with respect to a

Letk≥2 be a positive integer and letap∈m_E. In this section, we will apply the continuity properties of the Wach modules to prove local constancy results modulo prime powers when we fix the weightk and we let the trace of the crystalline Frobeniusa_p varyp-adically.

The main result of this section is the following (this is a generalization of Thm. A of [Ber12]):

Theorem 4.5. Let ap, a⁰_p ∈ m_E and k ≥ 2 be an integer. Let m ∈ ¹_e(Z≥1) such that v(ap−a⁰_p) ≥ 2·v(ap) +α(k−1) +m, then for every G_Qp-stable lattice Tk,a_p inside Vk,a_p there exists aG_Qp-stable latticeTk,a⁰_p insideVk,a⁰_p such that

Tk,a_p⊗_OEO_E/(p^m)∼=Tk,a⁰_p⊗_OEO_E/(p^m)asG_Qp-modules.

Proof. Consider the G_Qp-representation V_k,a^∗

p = Hom_E(Vk,a_p,E); it is crystalline and it has Hodge- Tate weights 0 and −(k−1). LetT_k,a^∗

p := HomO_E(Tk,a_p,O_E) be the G_Qp-stable lattice inV_k,a^∗

p, dual of Tk,ap. By a result of Berger (see prop. III.4.2 and III.4.4 in [Ber04]), it is possible to attach to T_k,a^∗

p a Wach moduleNk,a_p of heightk−1. Fixing a basis ofNk,a_p as A⁺

E-module (we recall that in our notation A⁺_E =O_E[[x]]), the actions of ϕ and γ on Nk,a_p can be respectively represented by the matricesP ∈Mat₂(A⁺

E) andQ∈Id +xMat₂(A⁺

E). Note that since the actions ofϕ and Γ commute (in a semi-linear sense) we have that P ϕ(G) = Gγ(P). We recall also that the matrix P(0) has characteristic polynomialT²−apT +p^k−1. Now, let a⁰_p ∈ O_E be as in the hypothesis, i.e. it satisfies v(ap−a⁰_p)≥2·v(ap) +α(k−1) +mfor somem∈ ¹_e(Z≥1). Applying in sequence the results in section 4.1, we deduce the existence of two matricesP⁰ andG⁰ which give rise to a Wach moduleN⁰ and such that P ≡ P⁰ modp^m and G ≡ G⁰ modp^m. Since by construction P⁰ = (Id +H)P, we have that evaluating atx= 0 we deduce that the characteristic polynomial ofP⁰(0) isT²−a⁰_pT+p^k−1(note that Trace(H(0)P(0)) =a⁰_p−ap and Det(Id +H(0)) = 1). By a result of Berger (see prop. 1.2 in [Ber12]), we can deduce thatN⁰=Nk,a⁰_p, or equivalentlyVk,a⁰_p=T(N⁰)⊗O_EEandDcris(V_k,a^∗ 0

p) =N⁰/xN⁰⊗O_EE. SinceP ≡P⁰ modp^mandG≡G⁰ modp^m, we have that Nk,a_p ≡Nk,a⁰_p modp^m. As a consequence of Remark 1, we have thatD(Nk, ap)≡D(Nk,a⁰_p) modp^m, i.e.

D(Nk, ap)⊗_OEO_E/(p^m)∼=D(Nk,a⁰_p)⊗_OEO_E/(p^m) in the categoryMod^´et_(ϕ,Γ)(O_E).

Hence, we define Tk,a⁰_p := T(D(Nk,a⁰_p)^∗) which is a G_Qp-stable lattice in Vk,a⁰_p that satisfies (since Fontaine’s functorTis compatible with duals):

T_k,ap⊗O_EO_E/(p^m)∼=T_k,a⁰

p⊗O_EO_E/(p^m) asG_Qp-modules.

Indeed, sinceD(N_k,ap)≡D(N_k,a⁰

p) mod p^m, by exactness of Fontaine’s functorTwe can deduce that T_k,a^∗

p≡T(D(N_k,a⁰

p)) modp^m. This completes the proof of the theorem.

4.3 Converse of local constancy with respect to the trace

Via the continuity properties of the Wach modules, it is also possible to find an explicit necessary condition for the existence of local constancy phenomena modulo prime powers. In precise terms, let k≥2 be an integer and letap, a⁰_p∈m_E; then we have the following:

Proposition 4.6. Let m ∈ ¹_e(Z≥1) and assume m ≥ α(k−1). If V_k,ap ≡ V_k,a⁰

p modp^m, then v(a_p−a⁰_p)≥m−α(k−1).

Proof. This is a straightforward application of Berger’s Proposition 3.1. Indeed, we have thatD_cris(V_k,a^∗

p) = Nk,a_p/xNk,a_p⊗EandDcris(V_k,a^∗ 0

p) =Nk,a⁰_p/xNk,a⁰_p⊗Efor some Wach modulesNk,a_p andNk,a⁰_p corre- sponding respectively to theG_Qp-stable lattices inVk,a_p andVk,a⁰_p that are congruent modulo p^m. By proposition 3.1, we have thatNk,a_p≡Nk,a⁰_p modp^{m−α(k−1)} and looking at the characteristic polyno- mials ofϕacting onNk,a_p/xNk,a_p andNk,a⁰_p/xNk,a_p the claim follows.

5 Local constancy with respect to the weight

In this section, we are going to prove a local constancy result for reductions modulo prime powers once we fix the trace of the crystalline Frobeniusap and we let the weightkvary.

In order to simplify the notation, we will say that two E-linear representations V and V⁰ of G_Qp :=

Gal(Qp/Q^p) are congruent modulo some prime power (i.e. V ≡V⁰ modπ_Eⁿ for somen∈Z≥1) if there existG_Qp-stable latticesT ⊂V andT⁰ ⊂V⁰ such that we have an isomorphism

T⊗_OEO_E/(πⁿ)∼=T⁰⊗_OEO_E/(πⁿ) ofG_Qp-modules.

Note that the above definition requires a bit of attention when used as it clearly doesn’t define an equivalence relation (in general, it is not transitive).

The main result of this section is the following:

Theorem 5.1. Let pbe an odd prime. Let ap ∈m_E− {0} for some finite extension E/Qp. Letk≥2 be an integer andm∈ ¹_e(Z≥1)be fixed. Assume that

k≥(3v(a_p) +m)·

1− p

(p−1)² −1

+ 1. (∗)

There exists an integer r = r(k, ap) ≥ 1 such that if k⁰ −k ∈ p^r+m(p−1)Z≥0 then Vk,a_p ≡ Vk⁰,a_p

modp^m

Remark 3. As it will be clear from the proof below, the condition in the hypothesis is not optimal in the sense that it can be replaced by the weaker condition given by the system:

(k≥3v(ap) +α(k−1) + 1 +m, k⁰≥3v(a_p) +α(k⁰−1) + 1 +m.

as in Berger’s result (see Thm. B in [Ber12]) whenm= 1/e. For the sake of simplicity, we just assume the stronger condition (∗) which has the advantage that it is explicit in the weight, doesn’t depend on the functionαand automatically holds for k⁰ if it holds forkassumingk⁰≥k.

The condition (∗) in the theorem can be deduced directly from the above conditions in the system by noticing thatα(k−1) =P

n≥1

_k−1

pⁿ⁻¹(p−1)

satisfies the inequalityα(k−1)≤^(k−1)p_(p−1)2.

Remark 4. Note that Theorem 5.1 and Theorem 4.5 can be applied in sequence (i.e. one can first deform the weight and then deform the trace) in order to have a local constancy result in which both the trace and the weight vary. This is possible because Theorem 4.5 is independent of the starting chosen lattice insideVk,a_p. Note that the order in which the theorems can be applied in sequence cannot be switched, as it is always necessary to keep track of the lattices involved in the congruences.

Remark 5. It could be interesting to consider the question of finding explicitly a radius for the local constancy in the weight. Partial results have been obtained by Bhattacharya (see [Bha18]). As already pointed out in the introduction, the above theorem can be seen as a converse (in a special crystalline case) of a non-published theorem of Winterberger, proven by Berger and Colmez as a consequence of a continuity property of the Sen periods (see [BC08]). Such result provides a connection between the local constancy radius of our theorem and the constantc(2,Q^p) of Berger and Colmez. To be precise, combining Theorem 5.1 above and Cor. 7.1.2 in [BC08] we get the upper bound on the radius for the local constancy in weightp^−(r+m)≤p^{−(bm2c−c(2,Qp))}.

In order to prove the theorem, the idea is to make use of Kedlaya’s theory of (ϕ,Γ)-modules of slope zero over the Robba ring (see [Ked04]) and to realize the representationsVk,a_p (forksuff. big) as trianguline representations in the sense of Colmez (see [Col08]). A theorem of Colmez will then ensure us that locally such representations vary in a continuous way, in the sense that they come in analytic families. We will make this precise in the next section. We refer the reader to [Ber11] for a nice summary on the theory of trianguline representations and its applications in arithmetic geometry.

LetR_E be the Robba ring with coefficients in E and for any multiplicative character δ : Q^×p → E^×, denote byR_E(δ) :=R_Ee_δ the (ϕ,Γ)-module (in the sense of Kedlaya, see [Ked04]) of rank one obtained

by defining the actionsϕ(eδ) =δ(p)eδ andγ(eδ) =δ(χ(γ))eδ for allγ∈Γ, whereχdenotes the chosen fixed isomorphism between Γ andZ^×p.

Colmez (see Thm. 0.2 in [Col08]) proved that all (ϕ,Γ)-modules of rank one arise as R_E(δ) for a unique multiplicative character δ; moreover, if δ1, δ2 : Q^×p → E^× are multiplicative characters then Ext¹(R_E(δ₁),R_E(δ₂)) is an E-vector space of dimension 1 unlessδ₁δ₂⁻¹ is of the form x⁻ⁱ for some in- tegeri≥0, or|x|xⁱfor some integeri≥1; in both cases, the dimension overEis two and the attached projective space is isomorphic toP¹(E); here xdenotes the identity character of Q^×p.

Hence, where the extension is not unique (up to isomorphism), one will need to specify the corresponding parameter inP¹(E) usually calledL-invariant and denoted as L. The corresponding Galois representa- tion will be denoted byV(δ1, δ2,L). For an extensive discussion aboutL-invariant, we refer the reader to the original article of Colmez (see sec. 4.5 in [Col08]).

Each trianguline representationV(δ1, δ2) corresponds (up to considering blow-up in case δ1δ⁻¹₂ =x⁻ⁱ or|x|xⁱ; see [Col08]) to the point (δ1, δ2)∈X×XwhereXis isomorphic (non-canonically, as there are choices involved) to the Qp-rigid analytic space µ(Qp)×G^rigm ×B¹(1,1)⁻_Q

p, which parametrizes multi- plicative charactersQ^×p with values in the multiplicative group of some finite extension ofQp.

From now on, we denote byB¹(a, r)⁺_Q

p the closed affinoid rigidQp-ball centered inaand with radiusr.

The expressionB¹(a, r)⁻

Qp will instead denote the open rigidQp-ball as aQp-rigid analytic space. If we are working with an algebraic closureQp, then the expressionB¹(a, r)⁺will simply denote the standard p-adic ball centered in awith radiusr.

5.1 Proof of Theorem 5.1

Letk⁰ be an integer satisfyingk⁰−k∈(p−1)Z≥0. The claim is to prove that ifk⁰andkare sufficiently p-adically close then the corresponding representations are isomorphic modulo a prescribed prime power.

The assumption (∗) on the weightkallow us, applying Theorem 4.5, to deduce that:

Vk,a_p+^pk−1

ap

≡Vk,a_p modp^m, Vk⁰,a_p+^{pk0 −1}

ap

≡Vk⁰,a_p modp^m.

Indeed, note that assumption (∗) implies thatk−1>2v(a_p) and hence in this specific case, the Theorem 4.5 can be applied both starting from Vk,a_p or starting fromV

k,a_p+^pk−1

ap

(same goes for k⁰) and hence this gives us a strong control over the lattices involved in the congruences. Therefore, this first step reduces the claim to prove that ifk⁰ andkare sufficientlyp-adically close (in the weight space) then we have the congruenceV

k,ap+^pk_ap⁻¹ ≡V

k⁰,ap+^pk_ap^{0 −1} modp^m.

The following proposition of Colmez (see prop. 3.1 in [Ber12] or see sec. 4.5 in [Col08]) allow us to realize the above representations as trianguline representations:

Proposition 5.2. Ifz∈m_E is a root ofz²−a_pz+p^k−1which satisfiesv(z)< k−1, then we have that V(µ_z, µ1

zχ^1−k,∞) =V_k,a^∗

p.

Hereµz:Q^×p →E^× is the character which satisfiesµz(p) =z andµz(Z^×p) = 1 andχ:Q^×p →E^× is the character which satisfiesχ(p) = 1 andχ(y) =y for ally∈Z^×p.

Hence, if k⁰ −1 ≥ k−1 > v(ap), we have that the crystalline representations V

k,ap+^pk−1

ap

and Vk⁰,a_p+^{pk0 −1}

ap

coincide respectively with the trianguline representationsV(µa_p, δk,a_p,∞) andV(µa_p, δk⁰,a_p,∞), whereδ_k,ap :=µ1

apχ^1−k andδ_k⁰_,ap=µ1

apχ^1−k0. Since the L-invariant is going to be∞ for all the tri- anguline representations involved, we will drop the notationV(·,·,∞) writing simplyV(·,·).

The trianguline representationsV(µ_ap, δ_k,ap) andV(µ_ap, δ_k⁰_,ap) define twoE-points, respectivelyu_k,ap= (µa_p, δk,a_p) anduk⁰,a_p= (µa_p, δk⁰,a_p), on the rigid analytic spaceX×X(see [Col08]) parametrizing cou- ples of multiplicative characters ofQ^×p with values inL^× whereLis some finite extension ofQ^p.