Higher Coleman Theory

TL;DR

This paper introduces local cohomology techniques to analyze the finite slope part of Shimura variety cohomology, leading to classicality, vanishing results, and the construction of eigenvarieties, with applications to Galois representations.

Contribution

It develops a new framework using local cohomology and stratification methods to study p-adic families and eigenvarieties for Shimura varieties, extending overconvergent modular forms.

Findings

Spectral sequence relating local and classical cohomology

Classicality and vanishing theorems established

New properties of Galois representations proved

Abstract

We develop local cohomology techniques to study the finite slope part of the coherent cohomology of Shimura varieties. The local cohomology groups we consider are a generalization of overconvergent modular forms, and they are defined by using a stratification on the Shimura variety obtained from the Bruhat stratification on a flag variety via the Hodge-Tate period map. We construct a spectral sequence from the local cohomologies to the classical cohomology and use it to obtain classicality and vanishing results. We also develop a theory of p-adic families and construct eigenvarieties. As an application, we prove some new properties of Galois representations arising from certain non-regular algebraic cuspidal automorphic representations.