Algebra & Number Theory

Extended eigenvarieties for overconvergent cohomology

Christian Johansson and James Newton

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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.

Article information

Algebra Number Theory, Volume 13, Number 1 (2019), 93-158.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Secondary: 11F80: Galois representations

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


Johansson, Christian; Newton, James. Extended eigenvarieties for overconvergent cohomology. Algebra Number Theory 13 (2019), no. 1, 93--158. doi:10.2140/ant.2019.13.93.

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