Recently, Andreatta, Iovita and Pilloni constructed spaces of overconvergent modular forms in
characteristic ,
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
over a number field which are split at all places above
. If
is
, then we
obtain a new construction of the extended eigencurve of Andreatta–Iovita–Pilloni. If
is an inner
form of
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
(and mixed
characteristic). When
is
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 onlineon 27 February 2021.