Overconvergent algebraic automorphic forms

TL;DR

This paper develops a comprehensive theory of overconvergent p-adic automorphic forms and eigenvarieties for certain reductive groups, revealing new phenomena like semi-classical forms and a hierarchy of interpolation spaces.

Contribution

It extends existing constructions to a broader class of groups and introduces a hierarchy of eigenvarieties based on finite slope conditions, utilizing advanced representation theory methods.

Findings

Introduction of semi-classical automorphic forms

Hierarchy of eigenvarieties for different slope conditions

Extension of overconvergent form theory to new groups

Abstract

I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions due to Buzzard, Chenevier and Yamagami for forms of GL(n). This leads to some new phenomena, including the appearance of intermediate spaces of "semi-classical" automorphic forms; this gives a hierarchy of interpolation spaces (eigenvarieties) interpolating classical automorphic forms satisfying different finite slope conditions (corresponding to a choice of parabolic subgroup of G at p). The construction of these spaces relies on methods of locally analytic representation theory, combined with the theory of compact operators on Banach modules.