Extended eigenvarieties for overconvergent cohomology

Abstract

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 ${GL}_2∕\mathbb{Q}$, 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.