Wach models and overconvergence of \'etale $(\varphi, \Gamma)$-modules

TL;DR

This paper provides a new proof that all étale $(,g)$-modules over a finite extension of $Q_p$ are overconvergent, and establishes an explicit lower bound for the overconvergence radius using Wach models.

Contribution

It offers an alternative proof of overconvergence for étale $(,g)$-modules and introduces a uniform lower bound for the overconvergence radius based on Wach models.

Findings

All étale $(,g)$-modules are overconvergent over finite extensions of $Q_p$

Explicit lower bound for the overconvergence radius established

Method uses Wach models in modulo $p^n$ Galois representations

Abstract

A classical result of Cherbonnier and Colmez says that all \'etale $(\varphi, \Gamma)$-modules are overconvergent. In this paper, we give another proof of this fact when the base field $K$ is a finite extension of $\mathbb Q_p$. Furthermore, we obtain an explicit ("uniform") lower bound for the overconvergence radius, which was previously not known. The method is similar to that in a previous joint paper with Tong Liu. Namely, we study Wach models (when $K$ is unramified) in modulo $p^n$ Galois representations, and use them to build an overconvergence basis.

Equations

$$\underline{\hat{M}}(T):=({\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}{\otimes }_{{\mathbb{Z}}_p}T{)}^{H_K}.$$

$${\underline{\hat{M}}}^{†,r}(T):=({\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}^{†,r}{\otimes }_{{\mathbb{Z}}_p}V{)}^{H_K}.$$

$$\underline{\hat{M}}(T)={\mathcal{O}}_{\mathcal{E}}{\otimes }_{{\mathcal{O}}_{\mathcal{E}}^{†,r}}{\underline{\hat{M}}}^{†,r}(T).$$

$$(0,\frac{1}{21pf{e}^2{d}^2{p}^{fed}}],$$

$$V(M):=({\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}{\otimes }_{{\mathcal{O}}_{{\mathcal{E}}_K}}M{)}^{\phi =1}.$$

$$\underline{M}(V):=({\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}{\otimes }_{{\mathbb{Z}}_p}V{)}^{H_K}.$$

$$V(\hat{M}):=({\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}{\otimes }_{{\mathcal{O}}_{{\mathcal{E}}_K}}M{)}^{\phi =1},$$

$$\underline{\hat{M}}(V):=({\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}{\otimes }_{{\mathbb{Z}}_p}V{)}^{H_K},$$

$$q:=\frac{\phi ([\underline{\epsilon }]-1)}{[\underline{\epsilon }]-1}=\frac{\phi (u)}{u}=\frac{(1+u{)}^p-1}{u}.$$

$$T^{\ast }(\mathfrak{M}):=(\mathfrak{M}{\otimes }_{\mathfrak{S}}{\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}{)}^{\phi =1}.$$

$${\hat{T}}^{\ast }(\hat{\mathfrak{M}}):=(\mathfrak{M}{\otimes }_{\mathfrak{S}}{\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}{)}^{\phi =1}.$$

$$v_R(u(\mathrm{mod}p))=v_R(1-\epsilon )=\frac{p}{p-1}.$$

$$W(\text{Fr}R{)}^{†,r}:=\left\{x=\sum _{i=0}^{\infty }{p}^i[x_i]\in W(\text{Fr}R)|i+r\frac{p-1}{p}{v}_R(x_i)\to +\infty \right\}.$$

$$A^{†,r}:=A\cap W(\text{Fr}R{)}^{†,r}.$$

$$\underline{\hat{M}}(T):=({\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}{\otimes }_{{\mathbb{Z}}_p}T{)}^{H_K}.$$

$${\underline{\hat{M}}}^{†,r}(T):=({\mathcal{O}}_{{\hat{\mathcal{E}}}^{\text{ur}}}^{†,r}{\otimes }_{{\mathbb{Z}}_p}T{)}^{H_K}.$$

$$\underline{\hat{M}}(T)={\mathcal{O}}_{\mathcal{E}}{\otimes }_{{\mathcal{O}}_{\mathcal{E}}^{†,r}}{\underline{\hat{M}}}^{†,r}(T).$$

$$(0,\frac{1}{21pf{e}^2{d}^2{p}^{fed}}].$$

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$$0\to p^{n-1}{T}_n\to Z_n\to {\tilde{L}}_{n-1}/p{\tilde{L}}_{n-1}\to 0.$$

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$$s:=p^{fd}+p-2$$

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$$M_i\stackrel{{\iota }_{i,j}}{⟶}{M}_j\stackrel{q_{j,i}}{⟶}{M}_i$$

$${\mathfrak{M}}^{i,j}:=\text{Ker}(p^i\mathfrak{M}\stackrel{p^{j-i}}{⟶}{p}^j\mathfrak{M}).$$

$$u^h{\mathfrak{M}}_{(n-1,1)}^{\prime }\subset {\mathfrak{M}}_{(n,1)}^{\prime }.$$

$${\tilde{L}}_n^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathring{\mathfrak{M}}}_{n,1}^{\prime }\subset {\mathfrak{M}}_{(n,1)}^{\prime }.$$

$${\tilde{L}}_n^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

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Full text

The author is partially supported by a postdoctoral position funded by Academy of Finland through Kari Vilonen.

1. 1 Introduction
2. 2 $(\varphi,\Gamma)$-modules, Wach modules, and overconvergence
3. 3 Torsion Wach models
4. 4 Overconvergent basis and main theorem

1. Introduction

1.1. The classical theorem of Cherbonnier-Colmez

Let us first set up some notations. Let $p$ be a prime. Let $k$ be a perfect field of characteristic $p$, $W(k)$ its ring of Witt vectors, $K_{0}:=W(k)[1/p]$, and $K/K_{0}$ a totally ramified finite extension with $e$ the ramification index. We fix an algebraic closure $\overline{K}$ of $K$ and set $G_{K}:=\operatorname{Gal}(\overline{K}/K)$. Define $\mu_{n}\in\overline{K}$ inductively such that $\mu_{1}$ is a primitive $p$-th root of unity and $(\mu_{n+1})^{p}=\mu_{n}$, and let $K_{p^{\infty}}:=\cup_{n=1}^{\infty}K(\mu_{n})$. Let $H_{K}:=\operatorname{Gal}(\overline{K}/K_{p^{\infty}})$, and $\Gamma_{K}:=\operatorname{Gal}(K_{p^{\infty}}/K)$.

Let $T$ be a finite free $\mathbb{Z}_{p}$-representation of $G_{K}$ of rank $d$. In [Fon90], for each such $T$ is associated an étale $(\varphi,\Gamma)$-module $\underline{\hat{M}}(T)$ where

[TABLE]

See §2 for any unfamiliar terms and more details. Here $\mathcal{O}_{\widehat{\mathcal{E}}^{\textnormal{ur}}}$ is a certain “period ring”. For the $\mathbb{Z}_{p}$-representation $T$, we can also define its “overconvergent periods” via:

[TABLE]

where $r\in\mathbb{R}^{>0}$, and $\mathcal{O}_{\widehat{\mathcal{E}}^{\textnormal{ur}}}^{\dagger,r}$ is the “overconvergent period ring”. We say that $\hat{M}(T)$ is overconvergent if it can be recovered by its “overconvergent periods”, i.e., if for some $r\in\mathbb{R}^{>0}$, we have

[TABLE]

The main theorem of [CC98] is the following:

1.1.1 Theorem** ([CC98]).**

For any finite free $\mathbb{Z}_{p}$-representation $T$ of $G_{K}$, its associated $(\varphi,\Gamma)$-module is overconvergent.

1.1.2 Remark*.*

The theorem was also reproved (and generalized to family version) by Berger and Colmez [BC08]. There is also a relatively more direct proof by Kedlaya [Ked15].

1.2. A reproof when $K/\mathbb{Q}_{p}$ is finite

In this paper, we give another proof of Theorem 1.1.1 when $K$ is a finite extension of $\mathbb{Q}_{p}$. Furthermore, we obtain an explicit “uniform” lower bound (depending only on $p,f,e,d$ where $f:=[k:\mathbb{F}_{p}]$) on the overconvergence radius, which was previously not known.

1.2.1 Theorem**.**

Suppose $K/\mathbb{Q}_{p}$ is a finite extension. Let $T$ be a finite free $\mathbb{Z}_{p}$-representation of $G_{K}$ of rank $d$. Then $\underline{\hat{M}}(T)$ is overconvergent on the interval

[TABLE]

i.e., (1.1.1) holds for $r={1}/{(21pfe^{2}d^{2}p^{fed})}$ (see §2.4 for our conventions).

1.2.2 Remark*.*

1. (1)

We need $[K:\mathbb{Q}_{p}]<\infty$ in order to apply the loose crystalline lifting results in [GL], see [GL, Rem. 1.1.2(1)] for some remarks on this condition. 2. (2)

There are some results concerning the overconvergence radius in [BC08, §4.2] (cf. [BC08, Lem. 4.2.5, Prop. 4.2.6]). We will show that using loc. cit., together with results from [GL] and [CL11], we can also prove a certain “uniform overconvergence” (but only implicitly); see §4.3 for more details.

1.2.3 Remark*.*

It is interesting to mention that in [GP], another proof of the overconvergence property of étale $(\varphi,\tau)$-modules is given (which works without assuming $[K:\mathbb{Q}_{p}]<\infty$); however, the proof makes full use of Thm. 1.1.1.

The overconvergence theorem 1.1.1 plays a fundamental role in the application of $(\varphi,\Gamma)$-modules to study various problems, see, e.g., [GL, §1.2] for some discussion. In particular, as we mentioned in loc. cit, overconvergence helps to link the category of all Galois representations to the category of geometric (i.e., semi-stable, crystalline) representations. Similarly as in [GL], we will already use such a link to prove Theorem 1.2.1. Namely, we use crystalline representations to “approximate” general Galois representations. By writing this paper, we hope that our approach can shed some more light on the deeper meaning of overconvergence. We also hope that this paper can serve as a useful companion to the paper [GL], so that the readers can compare between the setting of $(\varphi,\Gamma)$-modules and $(\varphi,\tau)$-modules.

1.3. Strategy of proof

As we mentioned earlier, the strategy is similar to [GL], with one particular caveat. Namely, for a lattice in a crystalline representation of $G_{K}$, we can always attach a $(\varphi,\hat{G})$-module by the main result of [Liu10], but we can not always attach a Wach module. Fortunately, when $K$ is unramified, i.e., when $K=K_{0}$, we can always attach a Wach module (we have to be careful here to avoid a “circular reasoning”, see Remark 2.2.4 for more details). Now, by an argument in [CC98], overconvergence of Galois representations is insensitive to inductions (of representations); thus it suffices to prove Theorem 1.2.1 when $K=K_{0}$, where we always have Wach modules (for crystalline representations). Then the strategy (indeed, the proof itself) will be parallel to [GL], except a few minor changes (in particular, the $\varphi$-action on period rings, cf. §2.3.5).

Let us give a quick sketch of the strategy here (very similar to [GL, §1.3]). We assume $K/\mathbb{Q}_{p}$ is finite and $K=K_{0}$ in this paragraph. By the loose crystalline lifting theorem in [GL], we can easily deduce that for each $n\geq 1$, $T_{n}:=T/p^{n}T$ admits a unique maximal liftable Wach model. Then we can analyse these models, and use them to build an overconvergence basis to prove the theorem.

1.4. Structure of the paper

In §2, we collect basic facts about étale $(\varphi,\Gamma)$-modules and Wach modules. We define what it means for an étale $(\varphi,\Gamma)$-module to be overconvergent, and state the theorem of Cherbonnier-Colmez. In §3, when $K/\mathbb{Q}_{p}$ is finite and $K=K_{0}$, we show the existence of Wach models, and analyse their various properties. Finally in §4, we build an overconvergence basis to prove the main theorem; we also make some comparison with known proofs of overconvergence.

1.5. Notations

1.5.1. Some notations in $p$-adic Hodge theory

We put $R:=\varprojlim\limits_{x\rightarrow x^{p}}\mathcal{O}_{\overline{K}}/p\mathcal{O}_{\overline{K}}$, equipped with its natural coordinate-wise action of $G_{K}$. Let $\textnormal{Fr}R$ be its fraction field. There is a unique surjective projection map $\theta:W(R)\rightarrow\widehat{\mathcal{O}}_{\overline{K}}$ to the $p$-adic completion $\widehat{\mathcal{O}}_{\overline{K}}$ of $\mathcal{O}_{\overline{K}}$, which lifts the projection $R\rightarrow\mathcal{O}_{\overline{K}}/p$ onto the first factor in the inverse limit. Let $A_{\textnormal{cris}}$, $B_{\textnormal{cris}}^{+}$, $B_{\textnormal{dR}}^{+}$ be the usual period rings.

Recall that we defined $\mu_{n}\in\overline{K}$ inductively such that $\mu_{1}$ is a primitive $p$-th root of unity and $(\mu_{n+1})^{p}=\mu_{n}$. Set $\underline{\varepsilon}:=(\mu_{i})_{i\geq 0}\in R$. Let $[\underline{\varepsilon}]\in W(R)$ be the Techmüller representative, and $t:=-\log([\underline{\varepsilon}])\in{A_{\textnormal{cris}}}$ as usual. Let $\mathfrak{S}=W[\![u]\!]$ with Frobenius extending the arithmetic Frobenius on $W(k)$ and $\varphi(u)=(u+1)^{p}-1$. We can embed the $W(k)$-algebra $W(k)[u]$ into $W(R)\subset{A_{\textnormal{cris}}}$ by the map $u\mapsto 1-[\underline{\varepsilon}]$. This embedding extends to an embedding $\mathfrak{S}\hookrightarrow W(R)$ which is compatible with Frobenious endomorphisms.

1.5.2. Fontaine modules and Hodge-Tate weights

When $V$ is a semi-stable representation of $G_{K}$, we let $D_{\textnormal{st}}(V):=(B_{\textnormal{st}}\otimes_{\mathbb{Q}_{p}}V^{\vee})$ where $V^{\vee}$ is the dual representation of $V$. The Hodge-Tate weights of $V$ are defined to be $i\in\mathbb{Z}$ such that $\textnormal{gr}^{i}D_{\textnormal{st}}(V)\neq 0$. For example, for the cyclotomic character $\varepsilon_{p}$, its Hodge-Tate weight is $\{1\}$.

1.5.3. Some other notations

Throughout this paper, we reserve $\varphi$ to denote Frobenius operator. We sometimes add subscripts to indicate on which object Frobenius is defined. For example, $\varphi_{\mathfrak{M}}$ is the Frobenius defined on $\mathfrak{M}$. We always drop these subscripts if no confusion will arise. Let $S$ be a ring endowed with Frobenius $\varphi_{S}$ and $M$ a module over $S$. We always denote $\varphi^{*}M:=S\otimes_{\varphi_{S},S}M$. Note that if $M$ has a $\varphi_{S}$-semi-linear endomorphism $\varphi_{M}:M\rightarrow M$ then $1\otimes\varphi_{M}:\varphi^{*}M\rightarrow M$ is an $S$-linear map. We also reserve $v$ to denote valuations which is normalized so that $v(p)=1$. Finally ${\rm M}_{d}(S)$ always denotes the ring of $d\times d$-matrices with entries in $S$ and $I_{d}$ denotes the $d\times d$-identity matrix.

Acknowledgement. I thank Laurent Berger and Tong Liu for some useful discussions. I also thank the anonymous referee(s) for helping to improve the exposition.

2. $(\varphi,\Gamma)$-modules, Wach modules, and overconvergence

In this section, except in §2.5 (the final subsection), we will let $K$ be as in §1.1, i.e., $K/\mathbb{Q}_{p}$ is not necessarily finite.

In this section, we first collect some basic facts on (integral and torsion) étale $\varphi$-modules, étale $(\varphi,\Gamma)$-modules, Wach modules and their attached representations. Then, we define what it means for an étale $(\varphi,\Gamma)$-module to be overconvergent, and state the classical overconvergence theorem.

2.1. Étale $\varphi$-modules and $(\varphi,\Gamma)$-modules

Let $\mathcal{O}_{\mathcal{E}}$ be the $p$-adic completion of $\mathfrak{S}[1/u]$. Our fixed embedding $\mathfrak{S}\hookrightarrow W(R)$ determined by ${\underline{\varepsilon}}$ uniquely extends to a $\varphi$-equivariant embedding $\iota:\mathcal{O}_{\mathcal{E}}\hookrightarrow W(\textnormal{Fr}R)$, and we identify $\mathcal{O}_{\mathcal{E}}$ with its image in $W(\textnormal{Fr}R)$. We note that $\mathcal{O}_{\mathcal{E}}$ is a complete discrete valuation ring with uniformizer $p$ and residue field $k(\!({\underline{\varepsilon}})\!)$ as a subfield of $\textnormal{Fr}R$. Let $\mathcal{E}$ denote the fractional field of $\mathcal{O}_{\mathcal{E}}$, $\mathcal{E}^{\textnormal{ur}}$ the maximal unramified extension of $\mathcal{E}$ inside $W(\textnormal{Fr}R)[\frac{1}{p}]$ and $\mathcal{O}_{\mathcal{E}^{\textnormal{ur}}}$ the ring of integers. Set $\mathcal{O}_{\widehat{\mathcal{E}}^{\textnormal{ur}}}$ the $p$-adic completion of $\mathcal{O}_{\mathcal{E}^{\textnormal{ur}}}$ and $\mathfrak{S}^{\textnormal{ur}}:=W(R)\cap\mathcal{O}_{\widehat{\mathcal{E}}^{\textnormal{ur}}}$.

Let $\mathcal{O}_{\mathcal{E}_{K}}:=(\mathcal{O}_{\widehat{\mathcal{E}}^{\textnormal{ur}}})^{H_{K}}$, and so $\Gamma_{K}$ acts on it. When $K=K_{0}$ (i.e., $K$ is unramified), we actually have $\mathcal{O}_{\mathcal{E}_{K}}=\mathcal{O}_{\mathcal{E}}$ (see e.g., first paragraph of [Ber04, §I.2]); in this case, we have $\gamma(u)=(1+u)^{\varepsilon_{p}(\gamma)}-1$ for $\gamma\in\Gamma_{K}$.

2.1.1 Definition**.**

Let ${}^{\prime}\textnormal{Mod}_{\mathcal{O}_{\mathcal{E}_{K}}}^{\varphi}$ denote the category of finite type $\mathcal{O}_{\mathcal{E}_{K}}$-modules $M$ equipped with a $\varphi_{\mathcal{O}_{\mathcal{E}_{K}}}$-semi-linear endomorphism $\varphi_{M}:M\rightarrow M$ such that $1\otimes\varphi:\varphi^{*}M\rightarrow M$ is an isomorphism. Morphisms in this category are just $\mathcal{O}_{\mathcal{E}_{K}}$-linear maps compatible with $\varphi$’s. We call objects in ${}^{\prime}\textnormal{Mod}_{\mathcal{O}_{\mathcal{E}_{K}}}^{\varphi}$ étale $\varphi$-modules.

2.1.2.

Let ${}^{\prime}\textnormal{Rep}_{\mathbb{Z}_{p}}(H_{K})$ (resp. ${}^{\prime}\textnormal{Rep}_{\mathbb{Z}_{p}}(G_{K})$ ) denote the category of finite type $\mathbb{Z}_{p}$-modules $T$ with a continuous $\mathbb{Z}_{p}$-linear $H_{K}$ (resp. $G_{K}$)-action. For $M$ in ${}^{\prime}\textnormal{Mod}_{\mathcal{O}_{\mathcal{E}_{K}}}^{\varphi}$, define

[TABLE]

For $V$ in ${}^{\prime}\textnormal{Rep}_{\mathbb{Z}_{p}}(H_{K})$, define

[TABLE]

2.1.3 Theorem** ([Fon90, Prop. A 1.2.6]).**

The functors $V$ and $\underline{M}$ induces an exact tensor equivalence between the categories ${}^{\prime}\textnormal{Mod}_{\mathcal{O}_{\mathcal{E}_{K}}}^{\varphi}$ and ${}^{\prime}\textnormal{Rep}_{\mathbb{Z}_{p}}(H_{K})$.

2.1.4 Definition**.**

An étale $(\varphi,\Gamma)$-module is a triple $(M,\varphi_{M},\Gamma_{M})$ where

• •

$(M,\varphi_{M})$ is an étale $\varphi$-module;

• •

$\Gamma_{M}$ is a continuous $\mathcal{O}_{\mathcal{E}_{K}}$-semi-linear $\Gamma_{K}$-action on $M$ which commutes with $\varphi_{M}$.

2.1.5 Convention**.**

Since we also use $M$ to denote an étale $\varphi$-module, we will now use $\hat{M}=(M,\varphi_{M},\Gamma_{M})$ to denote an étale $(\varphi,\Gamma)$-module. Clearly, this notation compares with the étale $(\varphi,\tau)$-modules that we use in [GL].

For an étale $(\varphi,\Gamma)$-module $\hat{M}=(M,\varphi_{M},\Gamma)$, define

[TABLE]

which is a $G_{K}$-representation. For $V$ in ${}^{\prime}\textnormal{Rep}_{\mathbb{Z}_{p}}(G_{K})$, define

[TABLE]

which is an étale $(\varphi,\Gamma)$-module.

2.1.6 Theorem** ([Fon90]).**

The functors $V$ and $\underline{\hat{M}}$ induces an exact tensor equivalence between the categories ${}^{\prime}\textnormal{Mod}_{\mathcal{O}_{\mathcal{E}_{K}}}^{\varphi,\Gamma}$ and ${}^{\prime}\textnormal{Rep}_{\mathbb{Z}_{p}}(G_{K})$.

2.2. Wach $\varphi$-modules and Wach modules

In this subsection, we assume $K=K_{0}$. As we mentioned earlier, we have $\mathcal{O}_{\mathcal{E}}=\mathcal{O}_{\mathcal{E}_{K}}$ in this case. We write

[TABLE]

2.2.1 Definition**.**

For a nonnegative integer $r$, we write ${}^{\prime}\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}$ for the category of finite-type $\mathfrak{S}$-modules $\mathfrak{M}$ equipped with a $\varphi_{\mathfrak{S}}$-semilinear endomorphism $\varphi_{\mathfrak{M}}:\mathfrak{M}\rightarrow\mathfrak{M}$ satisfying

• •

the cokernel of the linearization $1\otimes\varphi:\varphi^{*}\mathfrak{M}\rightarrow\mathfrak{M}$ is killed by $q^{r}$;

• •

the natural map $\mathfrak{M}\rightarrow\mathcal{O}_{\mathcal{E}}\otimes_{\mathfrak{S}}\mathfrak{M}$ is injective.

Morphisms in ${}^{\prime}\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}$ are $\varphi$-compatible $\mathfrak{S}$-module homomorphisms.

We call objects in ${}^{\prime}\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}$ Wach $\varphi$-module of height $r$. The category of finite free Wach $\varphi$-modules of height $r$, denoted $\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}$, is the full subcategory of ${}^{\prime}\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}$ consisting of those objects which are finite free over $\mathfrak{S}$. We call an object $\mathfrak{M}\in{{}^{\prime}\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}}$ a torsion Wach $\varphi$-module of height $r$ if $\mathfrak{M}$ is killed by $p^{n}$ for some $n$.

For any Wach $\varphi$-module $\mathfrak{M}\in^{\prime}\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}$, we define

[TABLE]

2.2.2 Definition**.**

A finite free (resp. torsion)Wach module of height $r$ is a triple $\hat{\mathfrak{M}}=(\mathfrak{M},\varphi_{\mathfrak{M}},\Gamma_{\mathfrak{M}})$, where

1. (1)

$(\mathfrak{M},\varphi_{\mathfrak{M}})\in{{}^{\prime}\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}}$ is a finite free (resp. torsion) Wach $\varphi$-module of height $r$; 2. (2)

$\Gamma_{\mathfrak{M}}$ is a continuous $\Gamma_{K}$-action on $\mathfrak{M}$ which commutes with $\varphi_{\mathfrak{M}}$; 3. (3)

$\Gamma_{\mathfrak{M}}$ acts on $\mathfrak{M}/u\mathfrak{M}$ trivially.

For any Wach module $\hat{\mathfrak{M}}$, we can attach a $\mathbb{Z}_{p}[G_{K}]$-module via

[TABLE]

2.2.3 Theorem**.**

[Ber04, KR09]** The functor $\hat{T}^{\ast}$ induces an anti-equivalence between the category of finite free Wach modules of height $r$ and the category of $G_{K}$-stable $\mathbb{Z}_{p}$-lattices in crystalline representations with Hodge-Tate weights in $\{-r,\dots,0\}$.

2.2.4 Remark*.*

The above theorem in this form is first proved by Berger [Ber04, Thm. 2], which critically uses results by Wach and Colmez [Wac96, Col99]. However, the work in [Col99] (which proves that crystalline representations are of finite height) uses the overconvergence theorem in [CC98].

Fortunately, there is a second proof of the above theorem by Kisin and Ren ([KR09, Thm. 0.1]), which is similar to [Kis06] (and does not use results in [CC98]). Note that both [Kis06] and [KR09] use Kedlaya’s result on slope filtration over the Robba ring ([Ked04]), but that is independent of [CC98].

In summary, by citing [KR09, Thm. 0.1] for Theorem 2.2.3, there would be no circular reasoning in our paper to reprove overconvergence theorem of [CC98].

2.2.5 Convention**.**

From now on in this paper, by a crystalline representation, we always mean a crystalline representation with non-positive Hodge-Tate weights.

2.2.6.

For any integer $s\geq 0$, let $\hat{\mathfrak{M}}(\varepsilon_{p}^{-s})$ denote the Wach $\varphi$-module corresponding to $\varepsilon_{p}^{-s}$ via our Theorem 2.2.3. It is easy to check that this is a rank-1 $\mathfrak{S}$-module with a base ${\mathfrak{e}}$ so that $\varphi({\mathfrak{e}})=q^{s}{\mathfrak{e}}$. For any Wach $\varphi$-module $\mathfrak{M}$, we denote $\mathfrak{M}(s):=\mathfrak{M}\otimes\mathfrak{M}(\varepsilon_{p}^{-s})$. So $\varphi_{\mathfrak{M}({s})}=q^{s}\varphi_{\mathfrak{M}}$.

2.3. Wach $\varphi$-models and Wach models

In this subsection, we assume $K=K_{0}$. As we mentioned earlier, we have $\mathcal{O}_{\mathcal{E}}=\mathcal{O}_{\mathcal{E}_{K}}$ in this case.

2.3.1 Definition**.**

1. (1)

Given an étale $\varphi$-module $M$ in ${}^{\prime}\textnormal{Mod}_{\mathcal{O}_{\mathcal{E}}}^{\varphi}$. If $\mathfrak{M}\in{{}^{\prime}\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}}$ is a Wach $\varphi$-module so that $M=\mathcal{O}_{\mathcal{E}}\otimes_{\mathfrak{S}}\mathfrak{M}$, then $\mathfrak{M}$ is called a Wach $\varphi$-model of $M$, or simply a model of $M$. 2. (2)

Given $\hat{M}:=(M,\varphi_{M},\Gamma_{M})$ a torsion (resp. finite free) $(\varphi,\Gamma)$-module. A torsion (resp. finite free) Wach module ${\hat{\mathfrak{M}}}:=(\mathfrak{M},\varphi_{\mathfrak{M}},\Gamma_{\mathfrak{M}})$ is called a model of $\hat{M}$ if $\mathfrak{M}$ is a model of $M$ and the isomorphism $\mathcal{O}_{\mathcal{E}}\otimes_{\mathfrak{S}}\mathfrak{M}\simeq M$ is compatible with $\Gamma_{K}$-actions on both sides.

The following lemma is obvious.

2.3.2 Lemma**.**

1. (1)

If $\mathfrak{M}$ is a model of $M$, then $T^{\ast}(\mathfrak{M})\simeq V(M)$ as $\mathbb{Z}_{p}[H_{K}]$-modules. 2. (2)

If ${\hat{\mathfrak{M}}}$ is a model of $\hat{M}$ then $\hat{T}^{\ast}({\hat{\mathfrak{M}}})\simeq V(\hat{M})$ as $\mathbb{Z}_{p}[G_{K}]$-modules.

Clearly, the most natural models of étale $(\varphi,\Gamma)$-modules come from lattices in crystalline representations.

2.3.3 Lemma**.**

Suppose $T$ is a $G_{K}$-stable $\mathbb{Z}_{p}$-lattice in a crystalline representation of $G_{K}$ with Hodge-Tate weights in $\{-r,\dots,0\}$. Let $\hat{M}:=(M,\varphi_{M},\Gamma_{M})$ be the $(\varphi,\Gamma)$-module associated to $T$. Let ${\hat{\mathfrak{M}}}:=(\mathfrak{M},\varphi_{\mathfrak{M}},\Gamma_{\mathfrak{M}})$ be the Wach-module associted to $T$. Then ${\hat{\mathfrak{M}}}$ is a model of $\hat{M}$.

Proof.

Easy. ∎

Now suppose that $T_{n}$ is a $p$-power torsion representation of $G_{K}$, and $M_{n}$ the étale $\varphi$-module associated to $T_{n}|_{H_{K}}$. A natural source of Wach $\varphi$-models of $M_{n}$ comes from the following. Suppose that we have a surjective map of $G_{K}$-representations $f:L\twoheadrightarrow T_{n}$ where $L$ is a crystalline finite free $\mathbb{Z}_{p}$-representation, then it induces the surjective map (which we still denote by $f$) $f:\mathcal{L}\twoheadrightarrow M_{n}$, where $\mathcal{L}$ is the étale $\varphi$-module associated to $L|_{H_{K}}$. If $\mathfrak{L}$ is the Wach $\varphi$-module associated to $L$, then by Lemma 2.3.3 above, $\mathfrak{L}$ is a Wach $\varphi$-model of $\mathcal{L}$. And so $f(\mathfrak{L})$ is clearly a Wach $\varphi$-model of $M_{n}$.

2.3.4 Example**.**

Let $T_{1}=\mathbb{F}_{p}$ be the trivial $G_{K}$-representation and $M_{1}$ denote the corresponding trivial étale $\varphi$-module. Then $\mathfrak{M}_{1}=k\llbracket u\rrbracket$ is a model of $M_{1}$ which is realized by the surjection $\mathbb{Z}_{p}\twoheadrightarrow\mathbb{F}_{p}$. We also have another surjection $\mathbb{Z}_{p}(1-p)\twoheadrightarrow\mathbb{F}_{p}$, which realizes another model $\mathfrak{M}_{1}(p-1)\subset M_{1}$. It is easy to check that $\mathfrak{M}_{1}(p-1)=u^{p-1}\mathfrak{M}_{1}$.

2.3.5. Comparison between Kisin modules and Wach $\varphi$-modules

Let us compare the following two situations:

1. (1)

(Situation 1.) Kisin modules (see, e.g., [GL, §2] for more details) for a general $K$ where $[K:K_{0}]=e$ (and let us fix a uniformizer $\pi$ of $K$, with Eisenstein polynomial $E(u)\in W(k)[u]$ of degree $e$). Note that $\varphi(u)=u^{p}$ in this case. 2. (2)

(Situation 2.) Wach $\varphi$-modules when $K=K_{0}$. Recall that $q=\frac{(1+u)^{p}-1}{u}\in W(k)[u]$ of degree $p-1$. Note that $\varphi(u)=(1+u)^{p}-1$ in this case.

The theories in these two situations have lots of similarities. In particular, when we prove something for Wach $\varphi$-modules, the proof could be almost verbatim as the corresponding result for Kisin modules; oftenly, we only need to change the $E(u)$ in Situation 1 to $q$ in Situation 2 (and some other slight modifications). We already see this in §2.2.6 (compare with [GL, Ex. 2.3.4]). More notably, when we are in modulo $p$ situations, note that $E(u)=u^{e}(\bmod p)$ and $q=u^{p-1}(\bmod p)$; then oftenly, we only need to change $e$ in Situation 1 to $p-1$ in Situation 2 to prove results about Wach modules. We already see this in Example 2.3.4 (compare with [GL, Ex. 2.4.4]). Here is another illustration of the general phenomenon (switch between $e$ and $p-1$):

• •

Given a $p$-torsion Kisin module in (Situation 1) of $E(u)$-height $r$ (cf. [GL, §2]), let $A$ be the matrix for $\varphi$, then there exists $B$ such that $AB=u^{er}$.

• •

Given a $p$-torsion Wach $\varphi$-module in (Situation 2) of $q$-height $r$, let $A$ be the matrix for $\varphi$, then there exists $B$ such that $AB=u^{(p-1)r}$.

2.4. Overconvergence

In this subsection, we let $K$ be as in §1.1.

For any $x\in W(\textnormal{Fr}R)$, we can write $x=\sum\limits_{i=0}^{\infty}p^{i}[x_{i}]$ with $x_{i}\in\textnormal{Fr}R$. Denote $v_{R}(\cdot)$ the valuation on $\textnormal{Fr}R$ and normalized by

[TABLE]

For any $r\in\mathbb{R}^{>0}$, set

[TABLE]

It turns out that $W(\textnormal{Fr}R)^{{\dagger},r}$ is a ring, stable under $G_{K}$-action but not Frobenius, namely $\varphi(W(\textnormal{Fr}R)^{{\dagger},r})=W(\textnormal{Fr}R)^{{\dagger},r/p}$. See [CC98, §II.1] for more details. For any subring $A\subset W(\textnormal{Fr}R)$, denote

[TABLE]

Recall that for a finite free $\mathbb{Z}_{p}$-representation $T$ of $G_{K}$, we can associate the $(\varphi,\Gamma)$-module via:

[TABLE]

Now for any $r>0$, we define

[TABLE]

2.4.1 Definition**.**

For a finite free $\mathbb{Z}_{p}$-representation $T$ of $G_{K}$, its associated $(\varphi,\Gamma)$-module is called overconvergent if there exists $r>0$ such that

[TABLE]

In particular, if above holds, we say that $\underline{\hat{M}}(T)$ is overconvergent on the interval $(0,r]$.

The main theorem of [CC98] is the following:

2.4.2 Theorem** ([CC98]).**

For any finite free $\mathbb{Z}_{p}$-representation $T$ of $G_{K}$, its associated étale $(\varphi,\Gamma)$-module is overconvergent.

The following lemma says that we can reduce the proof of Theorem 2.4.2 to the case when $K=K_{0}$.

2.4.3 Lemma**.**

To prove Theorem 2.4.2, it suffices to prove the cases when $K$ is unramified.

Proof.

Suppose we have already proved Theorem 2.4.2 when the base field is unramified. Now let $K$ be any field with $[K:K_{0}]=e$. Given a finite free $\mathbb{Z}_{p}$-representation $T$ of $G_{K}$ of rank $d$, then the induction $\operatorname{Ind}_{G_{K}}^{G_{K_{0}}}T$ is a finite free $\mathbb{Z}_{p}$-representation of $G_{K_{0}}$ of rank $ed$. By assumption, $\operatorname{Ind}_{G_{K}}^{G_{K_{0}}}T$ is overconvergent as a $G_{K_{0}}$-representation. By [CC98, Thm. II. 3.2.(i)] $\textnormal{Res}_{G_{K}}^{G_{K_{0}}}\operatorname{Ind}_{G_{K}}^{G_{K_{0}}}T$ is overconvergent as a $G_{K}$-representation. By Mackey decomposition, $\textnormal{Res}_{G_{K}}^{G_{K_{0}}}\operatorname{Ind}_{G_{K}}^{G_{K_{0}}}T$ contains $T$ as a direct summand, and so $T$ has to be overconvergent as a $G_{K}$-representation too. Note that if $\operatorname{Ind}_{G_{K}}^{G_{K_{0}}}T$ is overconvergent on the interval $(0,r]$ as a $G_{K_{0}}$-representation, then $T$ is also overconvergent on the interval $(0,r]$ as a $G_{K}$-representation ∎

2.5. A reproof when $K/\mathbb{Q}_{p}$ is finite extension

The main result of our paper is:

2.5.1 Theorem**.**

Suppose $K/\mathbb{Q}_{p}$ is a finite extension, and let $f:=[k:\mathbb{F}_{p}]$. Let $T$ be a finite free $\mathbb{Z}_{p}$-representation of $G_{K}$ of rank $d$. Then $\underline{\hat{M}}(T)$ is overconvergent on the interval

[TABLE]

By Lemma 2.4.3, it suffices to prove Thm. 2.5.1 when $K=K_{0}$. Then the main input will be Theorem 4.1.1 (which is analogue of [GL, Thm. 6.1.1]), which says that there exists an “overconvergent basis” of $\underline{\hat{M}}(T)$, with respect to which all entries of matrices for $\varphi$ and $\gamma\in\Gamma_{K}$ are overconvergent elements.

2.5.2 Remark*.*

1. (1)

As we can see from (2.5.1), we have obtained a “uniform” lower bound of overconvergence radius for all $p$-adic Galois representations, once we fix $p,e,f,d$. It is very possible that this uniform bound can have applications to situations where we consider a “family” of Galois representations (where $p,e,f,d$ are naturally fixed). 2. (2)

However, let us also mention that it seems almost impossible to use methods in this paper to reprove the overconvergence property of the family of $(\varphi,\Gamma)$-modules attached to a family of Galois representations (as in the setting of [BC08]); indeed, it seems impossible to construct and study loose crystalline liftings of a family of residual representations. (This does not contradict with the potential usefulness mentioned in Item 1 above.) 3. (3)

Actually, we only need to prove the $K=\mathbb{Q}_{p}$ case to deduce the full Theorem 2.5.1. So for example, we can consider inductions of the form $\operatorname{Ind}_{G_{K}}^{G_{\mathbb{Q}_{p}}}T$. This will not save much trouble for us (besides, we still use some finite unramified extensions in e.g. Lemma 3.1.8).

3. Torsion Wach models

In this section, we always assume $K/\mathbb{Q}_{p}$ is a finite extension and $K=K_{0}$. We study liftable Wach $\varphi$-models in torsion étale $\varphi$-modules.

3.1. Maximal Wach models and devissage

For $n\in\mathbb{Z}^{>0}$, suppose $T_{n}$ is a $p^{n}$-torsion representation of $G_{K}$ ($T_{n}$ is not necessarily free over $\mathbb{Z}/p^{n}\mathbb{Z}$). Let $M_{n}$ be the étale $\varphi$-module corresponding to $T_{n}|_{H_{K}}$. Recall that in the torsion case, a Wach $\varphi$-module $\mathfrak{M}\subset M_{n}$ is called a Wach $\varphi$-model if $\mathfrak{M}[\frac{1}{u}]=M_{n}$.

3.1.1 Definition**.**

A Wach $\varphi$-model $\mathfrak{M}$ is called loosely liftable (in short, liftable), if there exists a $G_{K}$-stable $\mathbb{Z}_{p}$-lattice $L$ inside a crystalline representation and surjective map $f:L\twoheadrightarrow T_{n}$ such that the corresponding map of étale $\varphi$-modules $f:\mathcal{L}\rightarrow M_{n}$ satisfies $f(\mathfrak{L})=\mathfrak{M}$, where $\mathcal{L}$ and $\mathfrak{L}$ are the étale $\varphi$-modules and Wach modules for $L$ (see the discussion before Example 2.3.4). In this case, we say that $\mathfrak{M}$ can be realized by the surjection $f:L\twoheadrightarrow T_{n}$.

3.1.2 Definition**.**

Let $E$ be a finite extension of $\mathbb{Q}_{p}$, $\mathcal{O}_{E}$ its ring of integers, $\varpi_{E}$ a uniformizer, $\mathfrak{m}_{E}$ the maximal ideal and $k_{E}=\mathcal{O}_{E}/\mathfrak{m}_{E}$ the residue field.

1. (1)

For $\bar{\rho}:G_{K}\rightarrow\textnormal{GL}_{d}(k_{E})$ a torsion representation, we say that a continuous representation $r:G_{K}\rightarrow\textnormal{GL}_{d}(\mathcal{O}_{E})$ is a strict $\mathcal{O}_{E}$-lift of $\bar{\rho}$ if $r(\bmod\mathfrak{m}_{E})\simeq\bar{\rho}$. 2. (2)

For a $p$-torsion representation of the form $\bar{\rho}:G_{K}\rightarrow\textnormal{GL}_{d}(k_{E})$, we say that “$E$ is big enough” (for $\bar{\rho}$) if each direct summand of $\bar{\rho}^{\textnormal{ss}}$ has strict crystalline $\mathcal{O}_{E}$-lift with non-positive Hodge-Tate weights.

3.1.3 Theorem**.**

Suppose $T_{n}$ is a $p^{n}$-torsion representation of $G_{K}$ ($T_{n}$ is not necessarily free over $\mathbb{Z}/p^{n}\mathbb{Z}$), then it admits loose crystalline lifts, i.e., there exists a $G_{K}$-stable $\mathbb{Z}_{p}$-lattice $L$ inside a crystalline representation and surjective map $f:L\twoheadrightarrow T_{n}$. Furthermore, if we let $T_{1}:=T_{n}/pT_{n}$ and suppose $E$ is big enough for $T_{1}\otimes_{\mathbb{F}_{p}}k_{E}$, then we can always make the loose crystalline lift to be finite free over $\mathcal{O}_{E}$.

Proof.

This is [GL, Thm 3.3.2]; let us sketch the proof here. We hope it can serve as a quick guide for readers not familiar with the proof of loc. cit.; also, we will use many of the (intermediate) results later in §3.2.

Step 1: $n=1$ case. Let $E$ be the degree $d$ unramified extension of $K_{0}$, where $d=\dim_{\mathbb{F}_{p}}T_{1}$; let $\mathcal{O}_{E}$ be the ring of integers and let $k_{E}$ be the residue field.

• •

By [GL, Lem. 3.1.3], $E$ is big enough for $T_{1}\otimes_{\mathbb{F}_{p}}k_{E}$ (cf. Def. 3.1.2).

• •

Then [GL, Thm. 3.2.1] shows that there exists a finite unramified extension ${K^{\prime}}$ of $K$, such that the restriction of $T_{1}\otimes_{\mathbb{F}_{p}}k_{E}$ to ${G_{{K^{\prime}}}}$ admits a strict crystalline $\mathcal{O}_{E}$-lift $\rho^{\prime}$. Then $L=\operatorname{Ind}^{G_{K}}_{G_{{K^{\prime}}}}\rho^{\prime}$ is a loose crystalline lift of $T_{1}$.

Step 2: induction argument. Now suppose our theorem is true for $n-1$, and consider the case $n$.

• •

Denote $T_{n-1}:=T_{n}/p^{n-1}T_{n}$. By induction hypothesis, let $f_{n-1}:\widetilde{L}_{n-1}\twoheadrightarrow T_{n-1}$ be a loose crystalline lift. Let $\breve{W}_{n}$ be the cartesian product of $\widetilde{L}_{n-1}\twoheadrightarrow T_{n-1}$ and $T_{n}\twoheadrightarrow T_{n-1}$. We have the following diagram of short exact sequences (of $\mathbb{Z}_{p}[G_{K}]$-modules).

[TABLE]

Let $Z_{n}:=\breve{W}_{n}/p\breve{W}_{n}$, then one readily checks that the following is short exact:

[TABLE]

• •

Let $E$ be the “big enough” field for the $p$-torsion representation $T_{n}/pT_{n}$ as in Step 1. Since $T_{n}/pT_{n}\twoheadrightarrow p^{n-1}T_{n}$, $E$ is also big enough for $p^{n-1}T_{n}\otimes_{\mathbb{F}_{p}}k_{E}$. By Step 1, there exists $K^{\prime\prime}/K$ finite unramified such that there exists a strict crystalline $\mathcal{O}_{E}$-lift $\widetilde{N}_{n}^{\prime\prime}\twoheadrightarrow p^{n-1}T_{n}\otimes_{\mathbb{F}_{p}}k_{E}|_{G_{K^{\prime\prime}}}$. Also, $\widetilde{L}_{n-1}$ is a strict crystalline $\mathcal{O}_{E}$-lift of $\widetilde{L}_{n-1}/\varpi_{E}\widetilde{L}_{n-1}$. Via (Statement B) in the proof of [GL, Thm. 3.2.1] (which constructs extensions of strict crystalline lifts), we can find some finite unramified extension $K^{\prime}/K^{\prime\prime}$ such that we have the following diagram of short exact sequences of $G_{K^{\prime}}$-representations:

[TABLE]

where the first row are strict crystalline $\mathcal{O}_{E}$-lifts of the second row, and the maps from the second row to the third row are compatible projections to chosen $\mathbb{F}_{p}$-direct summands. Here we can easily compute that we can choose

[TABLE]

where $d=\dim_{\mathbb{F}_{p}}T_{n}/pT_{n}$.

• •

Define $\widetilde{L}_{n}^{\prime}:=\widetilde{Z}_{n}^{\prime}\times_{Z_{n}}\breve{W}_{n}$, where $\widetilde{Z}_{n}^{\prime}\rightarrow{Z_{n}}$ comes from diagram (3.1.3). The $G_{K^{\prime}}$-representation $\widetilde{L}_{n}^{\prime}$ sits in the following diagram of short exact sequences:

[TABLE]

One easily sees that $\widetilde{L}_{n}^{\prime}$ is $\mathcal{O}_{E}$-finite free and crystalline. Furthermore $\widetilde{L}_{n}^{\prime}$ maps surjectively onto $\breve{W}_{n}|_{G_{K^{\prime}}}$, and so onto $T_{n}|_{G_{K^{\prime}}}$. Finally, $\widetilde{L}_{n}:=\operatorname{Ind}_{G_{K^{\prime}}}^{G_{K}}\widetilde{L}_{n}^{\prime}$ is the desired loose crystalline lift of $T_{n}$.

∎

3.1.4 Lemma**.**

1. (1)

For any $M_{n}$ (coming from $T_{n}|_{H_{K}}$), a liftable model $\mathfrak{M}\subset M_{n}$ exists. 2. (2)

The set of liftable models inside $M_{n}$ has a unique maximal object (with respect to the obvious inclusion relation). (We denote the maximal liftable model as $\mathfrak{M}_{(n)}$).

Proof.

Item (1) follows from Thm. 3.1.3. Item (2) follows from similar argument as in [GL, Lem. 4.1.2]; let us sketch the argument for the reader’s convenience. Denote the set of all liftable models inside $M_{n}$ as $\textnormal{LF}_{\mathfrak{S}}^{\infty}(M_{n})$. It is trivial to see that $\textnormal{LF}_{\mathfrak{S}}^{\infty}(M_{n})$ admits finite supremum. Then it suffices to show that there is an upper bound for length of any chain in $\textnormal{LF}_{\mathfrak{S}}^{\infty}(M_{n})$; this is the content of [CL09, Lem. 3.2.4], which we need to reprove in our Wach $\varphi$-module setting. The proof is verbatim, if we change all $E(u)$ (resp. $e$) in loc. cit. to $q$ (resp. $p-1$) (cf. §2.3.5). ∎

3.1.5.

Now let us introduce some notations. Let $T$ be a $\mathbb{Z}_{p}$-finite free $G_{K}$-representation, and set $T_{n}:=T/p^{n}T$. Let $M$ be the finite free étale $\varphi$-module corresponding to $T$, and set $M_{n}:=M/p^{n}M$. Denote the natural projection $q_{j,i}:M_{j}\twoheadrightarrow M_{i}$ for $i<j$ induced by modulo $p^{i}$. Recall that we use $\mathfrak{M}_{(j)}$ to denote the maximal liftable Wach $\varphi$-model of $M_{j}$. Set $\mathfrak{M}_{(j,i)}=q_{j,i}(\mathfrak{M}_{(j)})$.

For $i<j$, we denote $\iota_{i,j}:M_{i}\rightarrow M_{j}$ the injective map where for $x\in M_{i}$, we choose any lift $\hat{x}\in M_{j}$, and let $\iota_{i,j}(x)=p^{j-i}\hat{x}$. This is clearly well-defined, and we will use it to identify $M_{i}$ with $\iota_{i,j}(M_{i})=M_{j}[p^{i}]=p^{j-i}M_{j}$ (recall that the notation $M_{j}[p^{i}]$ denotes the $p^{i}$-torsion elements). The maps $\iota_{i,j}$ are clearly transitive; namely, $\iota_{i,j}\circ\iota_{j,k}=\iota_{i,k}$. Also, the composite

[TABLE]

is precisely the map $M_{i}\overset{\times p^{j-i}}{\longrightarrow}M_{i}$.

Let $\mathfrak{M}\in{{}^{\prime}\textnormal{Mod}_{\mathfrak{S}}^{\varphi,r}}$ be a torsion Wach $\varphi$-module such that $\mathfrak{M}[\frac{1}{u}]=M_{n}$. Following the discussion above [Liu07, Lem. 4.2.4], for each $0\leq i<j\leq n$, we define

[TABLE]

3.1.6 Lemma**.**

We use the above notations. In particular, for $i<j$, we identify $M_{i}$ with $M_{j}[p^{i}]$. Then we have:

1. (1)

$\mathfrak{M}_{(j)}[p^{i}]=\mathfrak{M}_{(i)}$* as Wach $\varphi$-models of $M_{i}$.* 2. (2)

We have $\mathfrak{M}_{(j)}^{i-1,i}=\mathfrak{M}_{(i,1)}$ for $j\geq i$.

Proof.

The proof is strictly verbatim as in [GL, Lem. 4.1.5, Cor. 4.1.6]; let us sketch the main ideas. Suppose $f:L\twoheadrightarrow T_{j}$ realizes $\mathfrak{M}_{(j)}$. Let $g:L\twoheadrightarrow T_{j}\twoheadrightarrow p^{i}T_{j}$ be the composite map where the second map is the $\times p^{i}$ map, and let $K:=\textnormal{Ker}g$. Then the induced loose crystalline lifting $K\twoheadrightarrow T_{j}[p^{i}]$ realizes $\mathfrak{M}_{(j)}[p^{i}]$, and thus $\mathfrak{M}_{(j)}[p^{i}]\subset\mathfrak{M}_{(i)}$. For the other direction, note that $\mathfrak{M}_{(i)}+\mathfrak{M}_{(j)}$ is a liftable model in $M_{j}$. So we have $\mathfrak{M}_{(i)}\subset\mathfrak{M}_{(j)}$, and so $\mathfrak{M}_{(i)}\subset\mathfrak{M}_{(j)}[p^{i}]$.

For Item (2), note $\mathfrak{M}_{(j)}^{i-1,i}=p^{i-1}(\mathfrak{M}_{(j)}[p^{i}])$, which equals to $p^{i-1}\mathfrak{M}_{(i)}$ (=$\mathfrak{M}_{(i,1)}$) by Item (1). ∎

3.1.7.

Suppose $T_{n}$ is a $p^{n}$-torsion representation of $G_{K}$, $M_{n}$ the corresponding étale $\varphi$-module. Suppose $K^{\prime}$ is a finite unramified extension of $K$, with residue field $k^{\prime}$ and ring of integers $W(k^{\prime})$. Then it is easy to see that $M^{\prime}_{n}:=W(k^{\prime})\otimes_{W(k)}M_{n}$ is the corresponding étale $\varphi$-module for $T_{n}|_{G_{K^{\prime}}}$.

3.1.8 Lemma**.**

Use notations as in 3.1.7, and use $\mathfrak{M}^{\prime}_{(n)}$ to denote the maximal liftable model of $M^{\prime}_{n}$. Then we have $\mathfrak{M}^{\prime}_{(n)}\simeq W(k^{\prime})\otimes_{W(k)}\mathfrak{M}_{(n)}$.

Proof.

This is the $(\varphi,\Gamma)$-analogue of [GL, Lem. 4.2.10]. Similarly as the beginning of [GL, §4.2], we need to set $K^{\prime}_{p^{\infty}}:=K^{\prime}K_{p^{\infty}}$. Since we have $K^{\prime}\cap K_{p^{\infty}}=K$ (because $K/\mathbb{Q}_{p}$ is unramified), we still have $G_{p^{\infty}}/G_{p^{\infty}}^{\prime}\simeq\operatorname{Gal}(K^{\prime}/K)$, where $G_{p^{\infty}}^{\prime}:=\operatorname{Gal}(\overline{K}/K^{\prime}_{p^{\infty}})$. Then all the argument in [GL, §4.2] carry over verbatim. ∎

3.2. Existence of $h$

3.2.1 Proposition**.**

Use notations in 3.1.5. Then there exists a constant $h$ only depending on $p$, $f$ and $d$ such that $u^{h}\mathfrak{M}_{(n-1,1)}\subset\mathfrak{M}_{(n,1)}$ for all $n\geq 1$. Consequently $u^{h}\mathfrak{M}_{(n)}^{i-1,i}\subset\mathfrak{M}_{(n)}^{i,i+1}$ for all $i=1,\dots,n-1$ by Lemma 3.1.6(2).

Proof.

The proof is basically the same as that for [GL, Prop. 5.2.1], except a few minor changes (mainly due to the difference of Frobenius actions, cf. §2.3.5). For the reader’s convenience, we give a sketch of the main arguments; in particular, we point out the changes in our situation.

By Lemma 3.1.8, for any $n$, if we let $M_{n}^{\prime}:=W(k^{\prime})\otimes_{W(k)}M_{n}$ (where $k^{\prime}/k$ is any finite extension), then the maximal liftable Kisin model of $M_{n}^{\prime}$ is $\mathfrak{M}^{\prime}_{(n)}\simeq W(k^{\prime})\otimes_{W(k)}\mathfrak{M}_{(n)}$. Thus, it is easy to see that $\mathfrak{M}^{\prime}_{(n,1)}=W(k^{\prime})\otimes_{W(k)}\mathfrak{M}_{(n,1)}$ for any $n$. So to prove our proposition, it suffices to show that

[TABLE]

We divide the following argument into two steps. In Step 1, we will construct another Kisin model (denoted as $\mathring{\mathfrak{M}}^{\prime}_{n,1}$) of $M_{1}^{\prime}$ such that $\mathring{\mathfrak{M}}^{\prime}_{n,1}\subset\mathfrak{M}^{\prime}_{(n,1)}$. Then in Step 2, we show that $u^{h}\mathfrak{M}^{\prime}_{(n-1,1)}\subset\mathring{\mathfrak{M}}^{\prime}_{n,1}.$

Step 1. The loose crystalline lift $\widetilde{L}^{\prime}_{n}\twoheadrightarrow\breve{W}_{n}|_{G_{K^{\prime}}}\twoheadrightarrow T_{n}|_{G_{K^{\prime}}}$ realizes a liftable Kisin model $\mathring{\mathfrak{M}}^{\prime}_{n}$ of $M^{\prime}_{n}$, and so $\mathring{\mathfrak{M}}^{\prime}_{n}\subset\mathfrak{M}_{(n)}^{\prime}$. The following composite of $G_{K^{\prime}}$-representations

[TABLE]

realizes a Kisin model $\mathring{\mathfrak{M}}^{\prime}_{n,1}$ in $M_{1}^{\prime}$, and we certainly have

[TABLE]

By some elementary diagram chasing (using (3.1.1), (3.1.5) and (3.1.3); cf. [GL, Prop. 5.2.1] for the chasing), the composite (3.2.1) is the same as

[TABLE]

And so (3.2.2) also realizes $\mathring{\mathfrak{M}}^{\prime}_{n,1}$.

Step 2. The last surjection of (3.2.2) is induced by the following composite of $G_{K}$-representations:

[TABLE]

We may assume that $\widetilde{L}_{n-1}\twoheadrightarrow T_{n-1}$ realize $\mathfrak{M}_{(n-1)}$; thus the composite (3.2.3) restricted to $G_{K^{\prime}}$ realizes $\mathfrak{M}^{\prime}_{(n-1,1)}$.

So in order to prove $u^{h}\mathfrak{M}^{\prime}_{(n-1,1)}\subset\mathfrak{M}^{\prime}_{(n,1)}$, it suffices to show

[TABLE]

And so it suffices to show that the cokernel of the following composite (which are maps of Wach modules corresponding to (3.2.2)) is killed by $u^{h}$

[TABLE]

By the Wach module analogue of [GL, Lem. 5.2.2] (the proof in the Wach module setting is strictly verbatim), the map $\widetilde{\mathfrak{L}}^{\prime}_{n}\rightarrow\widetilde{\mathfrak{Z}}_{n}^{\prime}$ is in fact surjective, so we only need to consider the cokernel of the following composite of maps:

[TABLE]

Denote the image of the composite (3.2.6) as $\mathring{\mathfrak{L}}^{\prime}_{n-1}$, which is contained in $\mathfrak{L}^{\prime}_{n-1}(s)/p\mathfrak{L}^{\prime}_{n-1}(s)$. So we can choose basis $e_{1},\ldots e_{m}$ of $\mathfrak{L}^{\prime}_{n-1}$, such that $\mathring{\mathfrak{L}}^{\prime}_{n-1}$ has a $k^{\prime}[\![u]\!]$-basis formed by $u^{a_{1}}\bar{e}_{1},\ldots u^{a_{m}}\bar{e}_{m}$, where

[TABLE]

Here, the expression of “$a_{i}$” in (3.2.7) is different from that in [GL] (below [GL, Eqn. (5.2.5)]). This is essentially because in loc. cit., [GL, Ex. 2.4.4] is (implicitly) used; whereas in our situation, we use our corresponding Ex. 2.3.4.

Finally, it suffices to bound $a_{i}$; the proof follows similar ideas as in [GL, Lem. 5.2.6].

Convention: In the remaining of the proof, all the representations that we consider are $G_{K^{\prime}}$-representations, and all Wach modules are over $\mathfrak{S}^{\prime}=W(k^{\prime})\otimes_{W(k)}\mathfrak{S}$. To be completely rigorous, we will need to restrict many representations from $G_{K}$ to $G_{K^{\prime}}$, and use prime notation over Wach modules (i.e., notations like $\mathfrak{M}^{\prime}$). For notational simplicity, we will drop these prime notations.

• •

Firstly, given any $p$-torsion Wach $\varphi$-module $\mathfrak{M}$ (over $k^{\prime}[\![u]\!]$), denote (similarly as in [GL, Def. 5.2.4])

[TABLE]

The invariant $\alpha(\mathfrak{M})$ is well-defined, and is additive with respect to short exact sequences (as in [GL, Lem. 5.2.5(1)]). Furthermore, if $L$ is a $G_{K^{\prime}}$-stable $\mathbb{Z}_{p}$-lattice in a crystalline representation $V$ with non-positive Hodge-Tate weights $\textnormal{HT}(V)$, and $\mathfrak{L}$ the corresponding Wach module, then by the proof of [Ber04, III. 3.1], we have

[TABLE]

Note that (3.2.8) is the Wach module analogue of [GL, Lem. 5.2.5(2)].

• •

Now, recall that $\mathring{\mathfrak{L}}_{n-1}$ has a $k^{\prime}[\![u]\!]$-basis formed by $u^{a_{1}}\bar{e}_{1},\ldots u^{a_{m}}\bar{e}_{m}$, where $\bar{e}_{1},\ldots\bar{e}_{m}$ is a $k^{\prime}[\![u]\!]$-basis of $\mathfrak{L}_{n-1}/p\mathfrak{L}_{n-1}$. We clearly have

[TABLE]

Since $b_{i}\leq\sum_{1\leq i\leq m}b_{i}$, so we only need to bound $\alpha(\mathring{\mathfrak{L}}_{n-1})-\alpha(\widetilde{\mathfrak{L}}_{n-1}(-s)/p\widetilde{\mathfrak{L}}_{n-1}(-s))$. By exactly the same (easy) argument as in [GL, Lem. 5.2.6] (which uses [GL, Lem. 5.2.2]), we have

[TABLE]

Note in the current paper, the $\mathbb{Q}_{p}$-dimension of $\widetilde{N}_{n}[\frac{1}{p}]$ (constructed in the first row of (3.1.3)) should be $fd^{2}$, and so

[TABLE]

And so we can choose any $h$ such that $h\leq s+fd^{2}(p^{fd}-2)$. So in our current paper, we can choose

[TABLE]

∎

4. Overconvergent basis and main theorem

In this section, we prove our main theorem; then we also make some comparison with known proofs.

4.1. Overconvergent basis

We first show the existence of an “overconvergent basis”, with respect to which all entries of the matrices for $\varphi$ and $\gamma\in\Gamma_{K}$ are overconvergent elements. In the following, we will use notations like $(e_{j})$ to mean a row vector $(e_{j})_{j=1}^{d}=(e_{1},\cdots,e_{d})$.

4.1.1 Theorem**.**

Suppose $K/\mathbb{Q}_{p}$ is finite extension and $K=K_{0}$. Let $T$ be a finite free $\mathbb{Z}_{p}$-representation of $G_{K}$ of rank $d$. Let $h=3fd^{2}p^{fd}$ be as in (3.2.9). Then there exists an $\mathcal{O}_{\mathcal{E}}$-basis $(e_{1},\dots,e_{d})$ of $\underline{\hat{M}}(T)$ such that,

• •

$\varphi(e_{j})=(e_{j})A$* with $A\in{\rm M}_{d}(\mathfrak{S}[\![\frac{p}{x}]\!])\subset{\rm M}_{d}(\mathfrak{S}[\![\frac{p}{u^{4h+3ph}}]\!])$ where $x=u^{4h}(\varphi(u)^{3h})$.*

• •

$\gamma(e_{j})=(e_{j})B$* with $B\in{\rm M}_{d}(\mathfrak{S}[\![\frac{p}{y}]\!])$ where $y=y_{\gamma}=(\varphi(u))^{4h}(\gamma(u))^{3h}$, for any $\gamma\in\Gamma_{K}$.*

Proof.

The proof follows exactly the same ideas as in [GL, Thm. 6.1.1]. Indeed, the construction of the basis $(e_{1},\dots,e_{d})$ is verbatim as Step 1, Step 2 and Step 3 of loc. cit. (which we sketch below for the reader’s convenience). We only need to modify Step 4 of loc. cit. in our situation.

Step 1. Generators of $\mathfrak{M}_{(n)}$. First of all, by induction on $n$, we construct a specific set of generators $\{{\mathfrak{e}}_{(n),j}^{(i)},1\leq i\leq n\}_{j=1}^{d}$ of $\mathfrak{M}_{(n)}$ such that $\{p^{i-1}{\mathfrak{e}}_{(n),j}^{(i)}\}_{j=1}^{d}$ forms a $k\llbracket u\rrbracket$-basis of $\mathfrak{M}_{(n)}^{i-1,i}$.

We choose $\{{\mathfrak{e}}_{(1),j}^{(1)}\}_{j=1}^{d}$ any $k\llbracket u\rrbracket$-basis of $\mathfrak{M}_{(1)}$. Suppose we have defined $\{{\mathfrak{e}}_{(n-1),j}^{(i)},1\leq i\leq n-1\}_{j=1}^{d}$ the generators for $\mathfrak{M}_{(n-1)}$. By Lem. 3.1.6(2), we can use the map $\iota_{n-1,n}:M_{n-1}\rightarrow M_{n}$ to identify $\mathfrak{M}_{(n-1)}^{i-1,i}$ with $\mathfrak{M}_{(n)}^{i-1,i}$ when $1\leq i\leq n-1$. Now define

• •

${\mathfrak{e}}_{(n),j}^{(i)}:=\iota_{n-1,n}({\mathfrak{e}}_{(n-1),j}^{(i)})$ for $1\leq i\leq n-1,1\leq j\leq d$, and

• •

choose any $\{{\mathfrak{e}}_{(n),j}^{(n)}\}_{j=1}^{d}$ in $\mathfrak{M}_{(n)}$, so that $(p^{n-1}{\mathfrak{e}}_{(n),j}^{(n)})$ is a $k\llbracket u\rrbracket$-basis of $\mathfrak{M}_{(n)}^{n-1,n}=p^{n-1}\mathfrak{M}_{(n)}$.

This finishes the inductive definition.

Step 2. Basis of $M_{n}$. With above, now we define basis for $M_{n}$. By [GL, Lem. 5.1.2(2)] (note that this lemma is about module theory over $\mathfrak{S}$; there is no $\varphi$ involved), for any $x\geq 1$, we can write

[TABLE]

with $\Lambda_{x-1}\in{\rm M}_{d}(\mathfrak{S})$ such that $u^{h}\Lambda_{x-1}^{-1}\in{\rm M}_{d}(\mathfrak{S})$ and $Z_{x-1}\in{\rm M}_{d}(\mathfrak{S}[\frac{p}{u^{2h}}])$. Let

[TABLE]

We can arrange that $Y_{n}\in{\rm M}_{d}(\frac{1}{u^{2(n-1)h}}\mathfrak{S})$, and $Y_{n}^{-1}\in{\rm M}_{d}(\frac{1}{u^{3(n-1)h}}\mathfrak{S})$ (because everything is $p^{n}$-torsion here), and we can easily check (by Nakayama Lemma) that $(e^{(n)}_{1},\ldots e^{(n)}_{d})$ is an $\mathcal{O}_{\mathcal{E},n}$-basis of $M_{n}$.

Step 3. Compatibility of basis. We can easily check that $(e^{(n-1)}_{j})=(e^{(n)}_{j}\mod p^{n-1})$. So now we can define $(e_{j}):=\lim_{n\rightarrow\infty}(e^{(n)}_{j})$, which is a basis of $M$.

Step 4. Matrices for $\varphi$ and $\gamma$. Now, to prove our theorem, similarly as in Step 4 of [GL, Thm. 6.1.1] (which used [GL, Lem. 6.1.3]), we can apply our Lemma 4.1.2(2), and so it suffices to show that (for any $\gamma\in\Gamma_{K}$),

• •

$\varphi(e_{j}^{(n)})=(e_{j}^{(n)})A_{n}$ with $A_{n}\in{\rm M}_{d}(\frac{1}{x^{(n-1)}}\mathfrak{S})$ where $x=u^{4h}(\varphi(u)^{3h})$.

• •

$\gamma(e_{j}^{(n)})=(e_{j}^{(n)})B_{n}$ with $B_{n}\in{\rm M}_{d}(\frac{1}{y^{(n-1)}}\mathfrak{S})$ where $y=(\varphi(u))^{4h}(\gamma(u))^{3h}$.

Since $\mathfrak{M}_{(n)}$ comes from a loose crystalline lift, we can write

• •

$\varphi({\mathfrak{e}}_{(n),j}^{(n)})=\sum_{i=1}^{n}({\mathfrak{e}}_{(n),j}^{(i)})P_{i}^{(1)}$, with $P_{i}^{(1)}\in{\rm M}_{d}(\mathfrak{S})$,

• •

$\gamma({\mathfrak{e}}_{(n),j}^{(n)})=\sum_{i=1}^{n}({\mathfrak{e}}_{(n),j}^{(i)})Q_{i}^{(1)}$, with $Q_{i}^{(1)}\in{\rm M}_{d}(\mathfrak{S})$.

By [GL, Lem. 5.1.2] (note that this lemma is about module theory over $\mathfrak{S}$; there is no $\varphi$ involved), for all $i<n$, we can write

[TABLE]

So we can write

• •

$\varphi({\mathfrak{e}}_{(n),j}^{(n)})=({\mathfrak{e}}_{(n),j}^{(n)})P_{n}$, with $P_{n}\in{\rm M}_{d}(\mathfrak{S}[\frac{p}{u^{2h}}])$,

• •

$\gamma({\mathfrak{e}}_{(n),j}^{(n)})=({\mathfrak{e}}_{(n),j}^{(n)})Q_{n}$, with $Q_{n}\in{\rm M}_{d}(\mathfrak{S}[\frac{p}{u^{2h}}])$.

Then we have

• •

$A_{n}=Y_{n}P_{n}\varphi(Y_{n}^{-1})$,

• •

$B_{n}=Y_{n}Q_{n}\gamma(Y_{n}^{-1})$,

and we can easily conclude. Finally, the containment ${\rm M}_{d}(\mathfrak{S}[\![\frac{p}{x}]\!])\subset{\rm M}_{d}(\mathfrak{S}[\![\frac{p}{u^{4h+3ph}}]\!])$ is via Lemma 4.1.3(1). ∎

4.1.2 Lemma**.**

Suppose $v\in W(R)-pW(R)$.

1. (1)

$W(R)[\![\frac{p}{v}]\!]\subset W(\textnormal{Fr}R)^{\dagger}$. 2. (2)

Suppose that $y_{n}=\frac{x_{n}}{v^{(n-1)}}\in W_{n}(\textnormal{Fr}R)$ such that $y_{n+1}\equiv y_{n}\mod p^{n}$ in $W_{n+1}(\textnormal{Fr}R)$. If $x_{n}\in\mathfrak{S}$ for all $n$, then $y_{n}$ converges to a $y\in\mathfrak{S}[\![\frac{p}{v}]\!]$.

Proof.

Item (1) is easy. Item (2) is similar to [GL, Lem. 6.1.3], which is a special case of the current lemma (with $v=u^{\alpha}$). ∎

4.1.3 Lemma**.**

Suppose $a\in\mathbb{Z}^{\geq 0},b,m\in\mathbb{Z}^{\geq 1}$, then we have

1. (1)

$\mathfrak{S}[\![\frac{p}{u^{a}\varphi(u^{b})}]\!]\subset\mathfrak{S}[\![\frac{p}{u^{a+bp}}]\!]$. 2. (2)

$p^{m}\varphi(u^{mb})=p^{m}u^{pmb}(\bmod p^{m+1}\mathfrak{S})$. 3. (3)

$\varphi(u^{mb})\in u^{pmb-pm}W(R)(\bmod p^{m+1}W(\textnormal{Fr}R)).$**

Proof.

Write $\varphi(u)=u^{p}+pc$ for some $c\in\mathfrak{S}$. Then for $x=\sum_{n\geq 0}s_{n}(\frac{p}{u^{a}\varphi(u^{b})})^{n}\in\mathfrak{S}[\![\frac{p}{u^{a}\varphi(u^{b})}]\!]$ where $s_{n}\in\mathfrak{S}$, we have

[TABLE]

Since $\frac{1}{(1+\frac{pc}{u^{p}})^{bn}}\in\mathfrak{S}[\![\frac{p}{u^{p}}]\!]\subset\mathfrak{S}[\![\frac{p}{u^{a+pb}}]\!]$, we can deduce (1). For (2) and (3), simply consider the expansion of $(u^{p}+pc)^{mb}$. ∎

4.2. Proof of Main theorem

First, we prove our main theorem when $K=K_{0}$.

4.2.1 Proposition**.**

Suppose $K/\mathbb{Q}_{p}$ is a finite extension and $K=K_{0}$. Let $T$ be a finite free $\mathbb{Z}_{p}$-representation of $G_{K}$ of rank $d$. Then $\underline{\hat{M}}(T)$ is overconvergent on the interval

[TABLE]

Proof.

The proof is similar to that of [GL, Thm. 6.2.1].

Let $(e_{1},\dots,e_{d})$ be the basis of $M$ as in Theorem 4.1.1. Let $(t_{1},\dots,t_{d})$ be any basis of $T$, and let $(e_{1},\dots,e_{d})=(t_{1},\dots,t_{d})X$ where $X\in{\rm M}_{d}(\mathcal{O}_{\widehat{\mathcal{E}}^{\textnormal{ur}}})$. Then we have $\varphi(X)=XA$. In order to prove the theorem, it suffices to show that $X\in{\rm M}_{d}(W(R)[\![\frac{p}{u^{\beta}}]\!])$ for

[TABLE]

To prove the above assertion, by Lemma 4.1.2, it suffices to show that $u^{(n-1)\beta}X_{n}\in{\rm M}_{d}(W_{n}(R))$ where $X_{n}:=X(\mod p^{n})$. We prove this by induction on $n$. The case $n=1$ is verbatim as in [GL, Thm. 6.2.1]. Suppose the claim is true when $n\leq m$, and let us consider the case $n=m+1$. Write $X=\sum\limits_{\ell=0}^{\infty}p^{\ell}X^{\prime}_{\ell}$ with $X^{\prime}_{\ell}\in{\rm M}_{d}([\textnormal{Fr}R])$ where $[\textnormal{Fr}R]$ is the set of Teichmüller lifts, so that we have $X_{m+1}=X_{m}+p^{m}X^{\prime}_{m}(\bmod p^{m+1})$. It suffices to show $u^{m\beta}X^{\prime}_{m}\in{\rm M}_{d}([R])$.

From $\varphi(X_{m+1})=X_{m+1}A_{m+1}(\bmod p^{m+1}W(\textnormal{Fr}R))$, we have

[TABLE]

Multiply both sides of (4.2.3) with $\varphi(u^{m\beta})$. Then we have

• •

$\varphi(u^{m\beta})\varphi(X_{m})\in{\rm M}_{d}(W(R))$ by induction hypothesis.

• •

The term

[TABLE]

where

• –

$\frac{\varphi(u^{m\beta})}{u^{(m-1)\beta+m(\beta-2)}}\in W(R)(\bmod p^{m+1}W(\textnormal{Fr}R)))$ by Lemma 4.1.3(3),

• –

$u^{(m-1)\beta}X_{m}\in{\rm M}_{d}(W(R))$ by induction hypothesis,

• –

and $u^{m(\beta-2)}A_{m+1}\in{\rm M}_{d}(W(R))$ by Theorem 4.1.1.

So the term $\varphi(u^{m\beta})X_{m}A_{m+1}\in{\rm M}_{d}(W(R))(\bmod p^{m+1}W(\textnormal{Fr}R)))$.

• •

The term $p^{m}\varphi(u^{m\beta})X_{m}^{\prime}A_{m+1}=p^{m}u^{m\beta}X_{m}^{\prime}u^{pm\beta-m\beta}A_{m+1}(\bmod p^{m+1}W(\textnormal{Fr}R)))$ by Lemma 4.1.3(2), and we have $u^{pm\beta-m\beta}A_{m+1}\in{\rm M}_{d}(W(R))$.

So in the end, we get an equation of the form

[TABLE]

where $Y=u^{m\beta}X^{\prime}_{m},B,C\in{\rm M}_{d}(W(R))$. We must have $C\in p^{m}{\rm M}_{d}(W(R))$, and so we can divide (4.2.4) by $p^{m}$ and apply [GL, Lem. 6.2.2] to conclude.

∎

We can deduce the full case from above:

4.2.2 Theorem**.**

Suppose $K/\mathbb{Q}_{p}$ is a finite extension. Let $T$ be a finite free $\mathbb{Z}_{p}$-representation of $G_{K}$ of rank $d$. Then $\underline{\hat{M}}(T)$ is overconvergent on the interval

[TABLE]

Proof.

This is easy corollary of Theorem 4.2.1, via Lemma 2.4.3. The expression of the overconvergence interval (4.2.5) is because the $\mathbb{Z}_{p}$-rank of the induction $\operatorname{Ind}_{G_{K}}^{G_{K_{0}}}T$ is $ed$. Indeed, (4.2.5) is an overconvergence interval for $\operatorname{Ind}_{G_{K}}^{G_{K_{0}}}T$ (as a $G_{K_{0}}$-representation), hence also for $T$ (as a $G_{K}$-representation). ∎

4.3. Comparison with known proofs

It is natural to wonder how to compare our proof with the classical proof of [CC98] (and also [BC08, Ked15]). However, there does not seem to be any obvious link between the two proofs. One particular characteristic of our proof is that the key technical analysis happen over the imperfect period rings (such that $\mathfrak{S}$ and $\mathcal{O}_{\mathcal{E}}$); whereas the classical proofs [CC98, BC08] (also implicitly in [Ked15]) rely on a Tate-Sen formalism (cf. [BC08, Def. 3.1.3]) for certain perfect period rings (cf., e.g., [BC08, Prop. 4.2.1]). The link between the two worlds remain mysterious.

As we already mentioned in the introduction, the explicit uniform bound on the overconvergence radius (4.2.5) is a new result. Actually, there are some results concerning the overconvergence radius in [BC08, §4.2] (we thank an anonymous referee for pointing out this reference). In the following, we show that using loc. cit., together with results from [GL] (on loose crystalline lifting) and [CL11] (on ramification bound), we can also prove a certain (implicit) “uniform overconvergence”.

In the following, let $K/\mathbb{Q}_{p}$ be a finite extension. Let $T$ be a finite free $\mathbb{Z}_{p}$-representation of $G_{K}$ of rank $d$. Suppose $L/K$ is a finite Galois extension such that $G_{L}$ acts trivially on $T/12pT$ (so a priori, $L$ depends on $T$, or at least on $T/12pT$).

Let $n(L)=n(G_{L})>0$ be the integer defined as in [BC08, Def. 3.1.3(TS3)], and let $s(L/K)>0$ be the number defined in [BC08, Lem. 4.2.5]. It is easy to see that both $n(L)$ and $s(L/K)$ depend only on $L$ and $K$. Now, let $r(L,K)=\min\{\frac{1}{(p-1)p^{n(L)-1}},\frac{1}{s(L/K)}\}$. Then by [BC08, Prop. 4.2.6], $\underline{\hat{M}}(T)$ is overconvergent on the interval $(0,r(L,K)]$ (note that the convention of $W(\textnormal{Fr}R)^{\dagger,r}$ in [BC08, §4.2] is different from ours in §2.4).

4.3.1 Lemma**.**

Suppose $p>2$. Then there exists a finite Galois extension $L=L(K,p,d)$ over $K$ which depends only on $K,p,d$, such that $G_{L}$ acts on $T/12pT$ trivially.

4.3.2 Remark*.*

By above lemma, when $p>2$, all the numbers $n(L)$, $s(L/K)$ and $r(L,K)$ depend only on $K,p,d$ (and not on $T$). So the $\underline{\hat{M}}(T)$’s are overconvergent “uniformly” (depending on $K,p,d$ only). However, in this method, we do not know how to explicitly bound the overconvergence radius, because it seems difficult to explicitly bound $n(L)$ and $s(L/K)$ (even though we can bound $L$ quite explicitly, see the proof in the following).

Proof of Lemma 4.3.1.

It suffices to show the existence of $L$ such that $G_{L}$ acts on $T/p^{2}T$ trivially. By [GL, Thm. 3.3.2, Rem. 3.3.5], there exists a loose crystalline lift of $T/p^{2}T$, with Hodge-Tate weights in the range $[-2(p^{fd}+p-2),0]$. Now we can apply [CL11, Thm. 1.1] to conclude. Namely, in loc. cit., we simply let $n=2$ and $r=2(p^{fd}+p-2)$. Clearly, the $s$ and $\mu$, and thus $G_{s}^{\mu}$ in loc. cit. depend on $K,p,d$ only. We then simply let our $L$ to be the Galois closure of the fixed field of $G_{s}^{\mu}$. ∎