Higher Coleman Theory

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.

References in corpus (2)

• Higher Hida and Coleman theories on the modular curve
• Coherent cohomology of Shimura varieties and automorphic forms

Cited by in corpus (5)

• On the Bloch--Kato conjecture for GSp(4) x GL(2)
• Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler--Shimura morphism
• Slope classicality via completed cohomology
• On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces
• Higher Hida theory for Hilbert modular varieties in the totally split case