Zariski Density of Crystalline Representations for Any $p$-Adic Field

J. Math. Sci. Univ. Tokyo
Vol. 21 (2014), No. 1, Page 79–127.

Nakamura, Kentaro
Zariski Density of Crystalline Representations for Any $p$-Adic Field
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The aim of this article is to prove Zariski density of crystalline representations in the rigid analytic space associated to the universal deformation ring of a $d$-dimensional mod $p$ representation of $\mathrm{Gal}(\overline{K}/K)$ for any $d$ and any $p$-adic field $K$. This is a generalization of the results of Colmez and Kisin for $d=2$ and $K=\mathbb{Q}_p$, of the author for $d=2$ and any $K$, and of Chenevier for any $d$ and $K=\mathbb{Q}_p$. A key ingredient for the proof is to construct a $p$-adic family of trianguline representations which can be seen as a local analogue of eigenvarieties. In this article, we construct such a family by generalizing Kisin's theory of finite slope subspace $X_{fs}$ for any $d$ and any $K$, and using Bella\"iche-Chenevier's idea of using exterior products in the study of trianguline deformations.

Keywords: $p$-adic Hodge theory, trianguline representations, B-pairs.

Mathematics Subject Classification (2010): Primary 11F80; Secondary 11F85.
Mathematical Reviews Number: MR3235551

Received: 2012-11-26