$S$-arithmetic (co)homology and $p$-adic automorphic forms

TL;DR

This paper investigates the $S$-arithmetic (co)homology of reductive groups over number fields, constructing eigenvarieties linked to parabolic subgroups and supercuspidal representations, and relating them to existing eigenvarieties.

Contribution

It introduces new constructions of eigenvarieties using $S$-arithmetic (co)homology and demonstrates their agreement with overconvergent homology methods and Bernstein eigenvarieties.

Findings

Eigenvarieties constructed from $S$-arithmetic (co)homology match those from overconvergent homology.

For definite unitary groups, the eigenvarieties relate closely to Bernstein eigenvarieties.

The approach extends the understanding of automorphic forms and their $p$-adic families.

Abstract

We study the $S$-arithmetic (co)homology of reductive groups over number fields with coefficients in (duals of) certain locally algebraic and locally analytic representations for finite sets of primes $S$. We use our results to construct eigenvarieties associated to parabolic subgroups at places in $S$ and certain classes of supercuspidal and algebraic representations of their Levi factors. We show that these agree with eigenvarieties constructed using overconvergent homology and that for definite unitary groups they are closely related to the Bernstein eigenvarieties constructed by Breuil-Ding.