Universal coefficients for overconvergent cohomology and the geometry of eigenvarieties

TL;DR

This paper establishes a universal coefficients theorem for overconvergent cohomology, enabling simpler construction of eigenvarieties and confirming conjectures about their dimensions, with specific results for certain reductive groups.

Contribution

It introduces a universal coefficients theorem for overconvergent cohomology and applies it to construct eigenvarieties and verify conjectures on their dimensions.

Findings

Eigenvarieties can be constructed using overconvergent cohomology.

Confirmed Urban's conjecture on eigenvariety dimensions in specific cases.

Cuspidal eigenvarieties for certain groups are rigid analytic curves.

Abstract

We prove a universal coefficients theorem for the overconvergent cohomology modules introduced by Ash and Stevens, and give several applications. In particular, we sketch a very simple construction of eigenvarieties using overconvergent cohomology and prove many instances of a conjecture of Urban on the dimensions of these spaces. For example, when the underlying reductive group is an inner form of GL(2) over a quadratic imaginary extension of the rationals, the cuspidal component of the eigenvariety is a rigid analytic curve.