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

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.