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

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 $(\phi ,\Gamma )$-modules over ${\tilde{\mathbb{A}}}_K^{+}$ 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 $(\phi ,\Gamma )$-modules over ${\tilde{\mathbb{A}}}_K$ that correspond to crystalline representations. This equivalence generalizes, in certain respects, that of L. Berger in the unramified case.

46 pages. Version 2: We corrected a mistake in Theorem 2.27 of version 1 (Theorem 2.30 in version 2). We also made minor changes. Version 3: We corrected an error concerning latex. Version 4: We slightly improve the main theorem. Along the way, subsection 2.2 and 2.3 are changed