Mathematical Research Letters

Volume 26 (2019)

Number 1

Irreducible components of extended eigenvarieties and interpolating Langlands functoriality

Pages: 159 – 201



Christian Johansson (Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Sweden)

James Newton (Department of Mathematics, King’s College London, United Kingdom)


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 $\mathrm{GL}_2 / F$, where $F / \mathbb{Q}$ is a cyclic extension, contains non-ordinary components of dimension at least $[F : \mathbb{Q}]$.

C.J. was supported by NSF grant DMS-1128155 and by the Herchel Smith Foundation.

Received 31 March 2017

Accepted 15 January 2018

Published 7 June 2019