Abstract:

Let $p$ be a prime, $K$ a finite extension of $\\Bbb\{Q\}_p$, and let $G_K$ be the absolute Galois group of $K$. The category of \\'etale $(\\varphi,\\tau)$-modules is equivalent to the category of $p$-adic Galois representations of $G_K$. In this paper, we show that all \\'etale $(\\varphi,\\tau)$-modules are overconvergent; this answers a question of Caruso. Our result is an analogue of the classical overconvergence result of Cherbonnier and Colmez in the setting of \\'etale $(\\varphi,\\Gamma)$-modules. However, our method is completely different from theirs. Indeed, we first show that all $p$-power-torsion representations admit loose crystalline lifts; this allows us to construct certain Kisin models in these torsion representations. We study the structure of these Kisin models, and use them to build an overconvergence basis.

摘要：

假设$ p $是素数，$ K $是$ \\ Bbb \ {Q \} _ p $的有限扩展，而$ G_K $是$ K $的绝对伽罗瓦群。\ e'tale $（\\ varphi，\\ tau）$-modules的类别等同于$ p $ -adic Galois表示形式的类别$ G_K $。在本文中，我们证明了所有\\'etale $（\\ varphi，\\ tau）$模块都是过度收敛的；这回答了卡鲁索的问题。我们的结果类似于Cherbonnier和Colmez在\ e'tale $（\\ varphi，\\ Gamma）$-modules设置中的经典超收敛结果。但是，我们的方法与他们的方法完全不同。确实，我们首先表明，所有$ p $ -power-torsion表示都承认松散的晶振；这使我们能够在这些扭转表示中构建某些Kisin模型。我们研究了这些Kisin模型的结构，并使用它们来构建超收敛基础。