Introduction

Part I. Background

A. Shimura varieties and automorphic vector bundles

  1. Examples and general definition
  2. Variation of Hodge structure, Deligne axioms, Riemann-Hilbert correspondence
  3. The Borel embedding and construction of automorphic vector bundles
  4. Harish-Chandra modules, Lie algebra cohomology, and Matsushima's formula

B. Hodge-Tate theory for modular curves

  1. Sen's theory
  2. Period rings and period sheaves
  3. A very quick survey of p-adic Hodge theory
  4. The perfectoid modular curve and Hodge-Tate period map
  5. Construction of automorphic vector bundles
  6. Fontaine's N operator

C. Elements of p-adic functional analysis

  1. p-adic Banach spaces and operators, representations of p-adic groups and locally analytic vectors
  2. Completed cohomology of the modular curve

D. Differential operators and localization

  1. Finite and infinite jet bundles
  2. Bruhat cells
  3. Riemann-Hilbert and Beilinson-Bernstein

Part II. Results

The remainder of the course will build on the background to explain some of the results obtained by Pan and the others on locally analytic vectors in completed cohomology and relations to modular forms and Galois representations.