On the $(\varphi,\Gamma)$-modules corresponding to crystalline representations

TL;DR

This paper defines crystalline $(, )$-modules over a certain ring and proves their category is equivalent to crystalline Galois representations, generalizing previous unramified case results.

Contribution

It introduces crystalline $(, )$-modules over $ ilde{A}_K^{+}$ and establishes an equivalence with crystalline Galois representations, extending Berger's unramified case work.

Findings

Crystalline $(, )$-modules are characterized over $ ilde{A}_K^{+}$.

The category of these modules is equivalent to crystalline Galois representations.

This generalizes Berger's unramified case results.

Abstract

Let $K$ be a complete discrete valuation field of characteristic $0$ with perfect residue field of characteristic $p>0$. We introduce the notion of crystalline $(\varphi,\Gamma)$-modules over $\widetilde{\mathbb{A}}_K^{+}$ and show that their category is equivalent to the category of crystalline $\mathbb{Z}_p$-representations of the absolute Galois group of $K$. In other words, we determine the $(\varphi,\Gamma)$-modules over $\widetilde{\mathbb{A}}_K$ that correspond to crystalline representations. This equivalence generalizes, in certain respects, that of L. Berger in the unramified case.