Sen Operators and Lie Algebras arising from Galois Representations over $p$-adic Varieties
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916567593254912 |
|---|---|
| author | He, Tongmu |
| author_facet | He, Tongmu |
| contents | Any finite-dimensional $p$-adic representation of the absolute Galois group of a $p$-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their construction to the fundamental group of a $p$-adic affine variety with a semi-stable chart, and prove that the module of Sen operators is canonically defined, independently of the choice of the chart. Our construction relies on a descent theorem in the $p$-adic Simpson correspondence developed by Tsuji. When the representation comes from a $\mathbb{Q}_p$-representation of a $p$-adic analytic group quotient of the fundamental group, we describe its Lie algebra action in terms of the Sen operators, which is a generalization of a result of Sen and Ohkubo. These Sen operators can be extended continuously to certain infinite-dimensional representations. As an application, we prove that the geometric Sen operators annihilate locally analytic vectors, generalizing a result of Pan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_07519 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sen Operators and Lie Algebras arising from Galois Representations over $p$-adic Varieties He, Tongmu Algebraic Geometry Number Theory 11F80 (primary), 14F35, 14G45 Any finite-dimensional $p$-adic representation of the absolute Galois group of a $p$-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their construction to the fundamental group of a $p$-adic affine variety with a semi-stable chart, and prove that the module of Sen operators is canonically defined, independently of the choice of the chart. Our construction relies on a descent theorem in the $p$-adic Simpson correspondence developed by Tsuji. When the representation comes from a $\mathbb{Q}_p$-representation of a $p$-adic analytic group quotient of the fundamental group, we describe its Lie algebra action in terms of the Sen operators, which is a generalization of a result of Sen and Ohkubo. These Sen operators can be extended continuously to certain infinite-dimensional representations. As an application, we prove that the geometric Sen operators annihilate locally analytic vectors, generalizing a result of Pan. |
| title | Sen Operators and Lie Algebras arising from Galois Representations over $p$-adic Varieties |
| topic | Algebraic Geometry Number Theory 11F80 (primary), 14F35, 14G45 |
| url | https://arxiv.org/abs/2208.07519 |