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Zariski density of crystalline representations for any $p$-adic field
Preprint 2011, arXiv:1104.1760v1 [math.NT]

Abstract. 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 $\operatorname{Gal}(\bar{K}/K)$ for any $d$ and for any $p$-adic field $K$.
This is a generalization of the results of Colmez, Kisin ($d=2$, $K=\mathbb{Q}_p$), of the author ($d=2$, any $K$), of Chenevier (any $d$, $K=\mathbb{Q}_p$). A key ingredient for the proof is to construct a $p$-adic family of trianguline representations. In this article, we construct (an approximation of) this family by generalizing Kisin's theory of finite slope subspace $X_{fs}$ for any $d$ and for any $K$.

Deformations of trianguline $B$-pairs and Zariski density of two dimensional crystalline representations
Preprint 2010, arXiv:1006.4891v1 [math.NT]

Abstract.The aim of this article is to study deformation theory of trianguline $B$-pairs for any $p$-adic field. For benign $B$-pairs, a special good class of trianguline $B$-pairs, we prove a main theorem concerning tangent spaces of these deformation spaces. These are generalizations of Bellaiche-Chenevier's and Chenevier's works in the $\mathbb{Q}_p$ case, where they used $(\phi,\Gamma)$-modules over the Robba ring instead of using $B$-pairs. As an application of this theory, in the final chapter, we prove a theorem concerning Zariski density of two dimensional crystalline representations for any $p$-adic field, which is a generalization of Colmez and Kisin's results in the $\mathbb{Q}_p$ case.

出版論文

2 次元クリスタリン表現のZariski 稠密性
RIMS Kôkyûroku Bessatsu B25: Algebraic Number Theory and Related Topics 2009, eds. T. Ichikawa, M. Kida, T. Yamazaki, June (2011), 161-184.
Classification of two dimensional split trianguline representations of $p$-adic fields
Compositio Mathematica, Volume 145 Part 4 (2009), pp. 865-914.
Available by subscription at Cambridge Journals Online.

Abstract. The aim of this article is to classify two-dimensional split trianguline representations of $p$-adic fields. This is a generalization of a result of Colmez who classified two-dimensional split trianguline representations of $\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ for $p\neq 2$ by using $(\varphi,\Gamma)$-modules over a Robba ring. In this article, for any prime $p$ and for any $p$-adic field K, we classify two-dimensional split trianguline representations of $\operatorname{Gal}(\overline{K}/K)$ using $B$-pairs as defined by Berger.


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