Extended eigenvarieties for overconvergent cohomology
Journal article, 2019

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.

p-adic automorphic forms

Galois representations


p-adic modular forms


Christian Johansson

Chalmers, Mathematical Sciences, Algebra and geometry

University of Gothenburg

James Newton

King's College London

Algebra and Number Theory

1937-0652 (ISSN)

Vol. 13 1 93-158

Subject Categories

Algebra and Logic


Mathematical Analysis


Basic sciences



More information

Latest update