#### Vol. 13, No. 1, 2019

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Extended eigenvarieties for overconvergent cohomology

### Christian Johansson and James Newton

Vol. 13 (2019), No. 1, 93–158
##### 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
Primary: 11F33
Secondary: 11F80

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