Loose crystalline lifts and overconvergence of étale $(\phi ,\tau )$-modules

Abstract

Let $p$ be a prime, $K$ a finite extension of ${\mathbb{Q}}_p$, and let $G_K$ be the absolute Galois group of $K$. The category of étale $(\phi ,\tau )$-modules is equivalent to the category of $p$-adic Galois representations of $G_K$. In this paper, we show that all étale $(\phi ,\tau )$-modules are overconvergent; this answers a question of Caruso. Our result is an analogy of the classical overconvergence result of Cherbonnier and Colmez in the setting of étale $(\phi ,\Gamma )$-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.

arXiv commentfinal version. to appear, Amer. J. Math