Irreducible components of extended eigenvarieties and interpolating Langlands functoriality

TL;DR

This paper investigates the geometry of extended eigenvarieties and establishes a broad interpolation theorem for Langlands functoriality, enhancing previous characteristic 0 results and revealing new geometric structures in characteristic p loci.

Contribution

It introduces a new geometric analysis of extended eigenvarieties and proves a general interpolation theorem for Langlands functoriality applicable to these spaces.

Findings

Characteristic p locus of extended eigenvariety contains non-ordinary components

Dimension of these components is at least the degree of the extension

Improves upon existing characteristic 0 interpolation results

Abstract

We study the basic geometry of a class of analytic adic spaces that arise in the study of the extended (or adic) eigenvarieties constructed by Andreatta--Iovita--Pilloni, Gulotta and the authors. We apply this to prove a general interpolation theorem for Langlands functoriality, which works for extended eigenvarieties and improves upon existing results in characteristic 0. As an application, we show that the characteristic p locus of the extended eigenvariety for GL(2)/F, where F is a cyclic extension of the rational numbers Q, contains non-ordinary components of dimension at least [F:Q].