## Deformations of Trianguline B-Pairs and Zariski Density of Two Dimensional Crystalline Representations

J. Math. Sci. Univ. Tokyo
Vol. 20 (2013), No. 4, Page 461–568.

Nakamura, Kentaro
Deformations of Trianguline B-Pairs and Zariski Density of Two Dimensional Crystalline Representations
The aims of this article are to study the deformation theory of trianguline $B$-pairs and to construct a $p$-adic family of two dimensional trianguline representations for any $p$-adic field. The deformation theory is the generalization of Bella\"iche-Chenevier's and Chenevier's works in the $\mathbb{Q}_p$-case, where they used ($\varphi,\Gamma$)-modules over the Robba ring instead of using $B$-pairs. Generalizing and modifying Kisin's theory of $X_{fs}$ for any $p$-adic field, we construct a $p$-adic family of two dimensional trianguline representations. As an application of these theories, we prove a theorem concerning the Zariski density of two dimensional crystalline representations for any $p$-adic field, which is a generalization of Colmez's and Kisin's theorem for the $\mathbb{Q}_p$-case.
Keywords: $p$-adic Hodge theory, trianguline representations, B-pairs.