Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the locally analytic completed cohomology of modular curves

Lue Pan

Abstract

We survey our works on the locally analytic vectors of completed cohomology of modular curves.

On the locally analytic completed cohomology of modular curves

Abstract

We survey our works on the locally analytic vectors of completed cohomology of modular curves.

Paper Structure

This paper contains 12 sections, 16 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Sen theory and localization of locally analytic completed cohomology
  3. Relative/Geometric Sen theory
  4. Localization: the sheaf $\mathcal{O}^{\mathrm{la}}$
  5. Explicit description of $\mathcal{O}^\mathrm{la}$
  6. Hodge-Tate structure: PanI
  7. Regular weights: $k\neq 1$
  8. Irregular weight: $k=1$
  9. de Rham structure: PanII
  10. Fontaine operator
  11. Geometric description of $N$
  12. Application

Key Result

Theorem 2.1

Let $X$ be a smooth rigid analytic variety over $C$ of dimension $d$ and $V$ a pro-étale $\mathbb{Z}_p$-local system on $X$. One can canonically define a map on the pro-étale site $X_{\mathrm{pro\acute{e}t}}$ where $\hat{\mathcal{O}}_X$ denotes the complete structural sheaf on $X_\mathrm{pro\acute{e}t}$ and $(-1)$ denotes the inverse of the Tate twist, satisfying the following

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.7
  • Definition 2.10
  • Theorem 2.11
  • Theorem 2.12
  • Theorem 2.13
  • Definition 2.18
  • Theorem 2.19
  • Theorem 3.1
  • Theorem 3.2
  • ...and 8 more