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arXiv · 2405.19829

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

Takumi Watanabe
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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.

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Takumi Watanabe. 2024-05-30. On the $(\varphi,\Gamma)$-modules corresponding to crystalline representations. https://arxiv.org/abs/2405.19829

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