Loose crystalline lifts and overconvergence of \'etale $(\varphi,   \tau)$-modules
  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 \'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 analogy 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. 
Hui Gao, Tong Liu
math.NT