Extended eigenvarieties for overconvergent cohomology

Johansson, Christian; Newton, James

doi:10.2140/ant.2019.13.93

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Extended eigenvarieties for overconvergent cohomology

Christian Johansson and James Newton

Vol. 13 (2019), No. 1, 93–158

DOI: 10.2140/ant.2019.13.93

Abstract

Recently, Andreatta, Iovita and Pilloni constructed spaces of overconvergent modular forms in characteristic $p$ , together with a natural extension of the Coleman–Mazur eigencurve over a compactified (adic) weight space. Similar ideas have also been used by Liu, Wan and Xiao to study the boundary of the eigencurve. This all goes back to an idea of Coleman.

In this article, we construct natural extensions of eigenvarieties for arbitrary reductive groups $G$ over a number field which are split at all places above $p$ . If $G$ is ${GL}_{2} ∕ ℚ$ , then we obtain a new construction of the extended eigencurve of Andreatta–Iovita–Pilloni. If $G$ is an inner form of ${GL}_{2}$ associated to a definite quaternion algebra, our work gives a new perspective on some of the results of Liu–Wan–Xiao.

We build our extended eigenvarieties following Hansen’s construction using overconvergent cohomology. One key ingredient is a definition of locally analytic distribution modules which permits coefficients of characteristic $p$ (and mixed characteristic). When $G$ is ${GL}_{n}$ over a totally real or CM number field, we also construct a family of Galois representations over the reduced extended eigenvariety.

A correction was submitted on 27 October 2020 and posted online on 27 February 2021.

Keywords

$p$-adic automorphic forms, $p$-adic modular forms, eigenvarieties, Galois representations

Mathematical Subject Classification 2010

Primary: 11F33

Secondary: 11F80

Supplementary material

Correction

Milestones

Received: 15 June 2017

Revised: 28 June 2018

Accepted: 25 September 2018

Published: 13 February 2019

Correction posted: 27 February 2021

Authors

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

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

Vol. 13, No. 1, 2019