Loose crystalline lifts and overconvergence of Étale (ϕ,τ)-modules

Let p be a prime, K a finite extension of Q,and let G be the absolute Galois group of K. The category of étale (ϕ,τ)-modules is equivalent to the category of p-adic Galois representations of G . In this paper, we show that all étale (ϕ,τ)-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 étale (ϕ,Γ)-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.