Hecke operators and the coherent cohomology of Shimura varieties

TL;DR

This paper investigates how Hecke operators act on the coherent cohomology of Shimura varieties, proposing a conjecture on their integral action and providing p-adic estimates for automorphic representations.

Contribution

It formulates a general conjecture on the integral action of Hecke operators and verifies it in specific cases, advancing understanding of automorphic forms and Shimura varieties.

Findings

Conjecture on integral Hecke action formulated

Verified in certain cases for specific Shimura varieties

Derived p-adic estimates for Satake parameters

Abstract

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain $p$-adic estimates of Satake parameters of certain non-regular self dual automorphic representations of $\mathrm{GL}_n$.