Parabolic eigenvarieties via overconvergent cohomology

TL;DR

This paper develops a new framework for overconvergent cohomology associated with parabolic subgroups, enabling the construction of $p$-adic eigenvarieties that interpolate systems of Hecke eigenvalues with controlled slopes.

Contribution

It introduces parahoric overconvergent cohomology groups, proves a classicality theorem with improved slope bounds, and constructs $Q$-parabolic eigenvarieties for broader arithmetic applications.

Findings

Classicality theorem with stronger slope bounds.

Construction of $Q$-parabolic eigenvarieties.

Overconvergent cohomology at parahoric level enhances flexibility.

Abstract

Let $G'$ be a connected reductive group over $\mathbb{Q}$ such that $G = G'/\mathbb{Q}_p$ is quasi-split, and let $Q \subset G$ be a parabolic subgroup. We introduce parahoric overconvergent cohomology groups with respect to $Q$, and prove a classicality theorem showing that the small slope parts of these groups coincide with those of classical cohomology. This allows the use of overconvergent cohomology at parahoric, rather than Iwahoric, level, and provides flexible lifting theorems that appear to be particularly well-adapted to arithmetic applications. When $Q$ is the Borel, we recover the usual theory of overconvergent cohomology, and our classicality theorem gives a stronger slope bound than in the existing literature. We use our theory to construct $Q$-parabolic eigenvarieties, which parametrise $p$-adic families of systems of Hecke eigenvalues that are finite slope at $Q$, but that allow infinite slope away from $Q$.