  Recently, Andreatta, Iovita and Pilloni have 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.

Number Theory