Eigenvarieties for cuspforms over PEL type Shimura varieties with dense ordinary locus

TL;DR

This paper constructs eigenvarieties for overconvergent cuspidal forms on PEL Shimura varieties with dense ordinary locus, demonstrating deformation of eigenvalues over weight space and defining families of p-adic modular forms.

Contribution

It introduces a geometric framework for families of overconvergent p-adic modular forms on PEL Shimura varieties and constructs eigenvarieties parametrizing eigenvalues of finite slope forms.

Findings

Eigenvarieties of expected dimension are constructed.

Finite slope eigenforms' eigenvalues deform over weight space.

Families of overconvergent forms are systematically defined.

Abstract

Let p>2 be a prime and let X be a compactified PEL Shimura variety of type (A) or (C) such that p is an unramified prime for the PEL datum. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens we define the notion of families of overconvergent locally analytic p-adic modular forms of Iwahoric level for X. We show that the system of eigenvalues of any finite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected dimension that parametrize systems of eigenvalues appearing in the space of families of cuspidal forms.