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Loose crystalline lifts and overconvergence of étale (φ, τ)-modules
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 142, Number 6, December 2020
- pp. 1733-1770
- 10.1353/ajm.2020.0043
- Article
- Additional Information
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.