A derived construction of eigenvarieties

TL;DR

This paper develops a derived version of eigenvarieties using advanced $p$-adic representation theory, comparing complexes, and establishing new exactness and representation results, with applications to eigenvarieties for $ ext{GL}_n$ over CM fields.

Contribution

It introduces a derived construction of eigenvarieties, compares homotopy equivalent complexes, and connects eigenvarieties of $ ext{GL}_n$ over CM fields to unitary groups.

Findings

Constructed a derived eigenvariety using homotopy equivalences.

Proved exactness of the finite slope part functor.

Established a subeigenvariety relationship for $ ext{GL}_n$ over CM fields.

Abstract

We construct a derived variant of Emerton's eigenvarieties using the locally analytic representation theory of $p$-adic groups. The main innovations include comparison and exploitation of two homotopy equivalent completed complexes associated to the locally symmetric spaces of a quasi-split reductive group $\mathbb{G}$, comparison to overconvergent cohomology, proving exactness of finite slope part functor, together with some representation-theoretic statements. As a global application, we exhibit an eigenvariety coming from data of $\mathrm{GL}_n$ over a CM field as a subeigenvariety for a quasi-split unitary group.