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2019 Extended eigenvarieties for overconvergent cohomology
Christian Johansson, James Newton
Algebra Number Theory 13(1): 93-158 (2019). DOI: 10.2140/ant.2019.13.93


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 GL2, then we obtain a new construction of the extended eigencurve of Andreatta–Iovita–Pilloni. If G is an inner form of GL2 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 GLn over a totally real or CM number field, we also construct a family of Galois representations over the reduced extended eigenvariety.


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Christian Johansson. James Newton. "Extended eigenvarieties for overconvergent cohomology." Algebra Number Theory 13 (1) 93 - 158, 2019.


Received: 15 June 2017; Revised: 28 June 2018; Accepted: 25 September 2018; Published: 2019
First available in Project Euclid: 27 March 2019

zbMATH: 07041707
MathSciNet: MR3917916
Digital Object Identifier: 10.2140/ant.2019.13.93

Primary: 11F33
Secondary: 11F80

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.13 • No. 1 • 2019
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