Eigenvarieties for non-cuspidal modular forms over certain PEL Shimura varieties

TL;DR

This paper constructs eigenvarieties for non-cuspidal automorphic forms on certain PEL Shimura varieties, extending previous methods to include forms with varying degrees of cuspidality and analyzing their dimensions.

Contribution

It generalizes the construction of eigenvarieties to non-cuspidal forms by gluing intermediate eigenvarieties, providing explicit dimension formulas and proving a conjecture for GSp4.

Findings

Eigenvarieties are constructed for non-cuspidal forms with explicit dimensions.

The dimension varies with the degree of cuspidality, being maximal for cuspidal forms.

Proved a conjecture of Urban regarding the dimension of eigenvarieties for GSp4.

Abstract

Generalising the recent method of Andreatta, Iovita, and Pilloni for cuspidal forms, we construct an eigenvariety for symplectic and unitary groups that parametrises systems of eigenvalues of overconvergent and locally analytic $p$-adic automorphic forms. This is achieved by gluing some intermediates eigenvarieties of a fixed 'degree of cuspidality'. The dimension of these eigenvarieties is explicit and depends on the degree of cuspidality, it is maximal for cuspidal forms and it is $1$ for forms that are 'not cuspidal at all'. Under mild assumption, we are able to prove a conjecture of Urban about the dimension of the irreducible components of Hansen's eigenvariety in the case of the group $\mathrm{GSp}_4$ over $\mathbb{Q}$.