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## プレプリント

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.

## 口頭発表

### 2011年度

• Zariski density of crystalline representations for any $p$-adic field, Boston-Keio Summer Workshop、September 17, 2011
• Zariski density of crystalline representations for any $p$-adic field
• Zariski density of crystalline representations for any $p$-adic field
代数セミナー、慶應義塾大学、April 21, 2011

### 2010年度

• Zariski density of crystalline representations
数論幾何セミナー、北海道大学、March 7, 2011
• Zariski density of crystalline representations
九州代数数的整数論2011、九州大学、February 17, 2011
• $GL_2(\mathbb{Q}_p)$の $p$進局所ラングランズ対応について
城崎新人セミナー、城崎、February 15, 2011
• Zariski density of crystalline representations
第9回仙台広島整数論集会、東北大学、July 22, 2010
• Construction of $p$-adic families of trianguline representations
玉原数論幾何研究集会、玉原、May 24, 2010

### 2009年度

• Trianguline表現と二次元クリスタリン表現のザリスキー稠密性について
数論幾何セミナー、名古屋大学、January 12, 2010
• 二次元クリスタリン表現の Zariski 稠密性
RIMS研究集会「代数的整数論とその周辺」2009, 東京大学, December 8, 2009
• Trianguline representations and Zariski density of two dimensional cristalline representations
Keio-Yonsei Cooperative Research Conference on Mathematical Sciences 2009, Yonsei University, November 19, 2009
• Zariski density of two dimensional trianguline representations of $p$-adic fields
Workshop: $p$-進特殊関数と数論幾何, October 29, 2009
• Zariski density of two dimensional cristalline representations
代数セミナー、九州大学、October 22, 2009
• Trianguline表現の分類とZariski稠密性について
日本数学会2009年度秋季総合分科会、代数分科会・一般講演、大阪大学、September 25, 2009
• $p$-進表現入門 Ⅰ、Ⅱ
第17回（2009年度）整数論サマースクール「l-進ガロア表現とガロア表現の整数論」、アピカルイン京都、August 17-21, 2009
• Zariski density of two dimensional trianguline representations of $p$-adic fields
$p$-adic Automorphic Forms and Arithmetic Geometry、気仙沼大島、July 28-31, 2009
• $p$-進体の二次元三角表現のZariski稠密性
第８回広島整数論集会、広島大学、July 24, 2009
• Zariski density of two dimensional trianguline representations of $p$-adic fields
代数セミナー、慶應義塾大学、June 18, 2009
• Zariski density of two dimensional trianguline representations of $p$-adic fields
大阪大学整数論 ＆ 保型形式セミナー、大阪大学、June 5, 2009

### 2008年度

• Classification of two dimensional split trianguline representations of $p$-adic fields
Industrious Number Theory、KIAS ソウル、March 11, 2009