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Eigenvarieties for Cuspforms Over PELType Shimura Varieties With Dense Ordinary Locus

Published online by Cambridge University Press:  20 November 2018

Riccardo Brasca
Riccardo Brasca*
Affiliation:
Institut de Mathématiques de Jussieu - Paris Rive Gauche, Université Paris Diderot, Paris, France e-mail: riccardo.brasca@gmail.com
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Abstract

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Let $p\,>\,2$ be a prime and let $X$ be a compactified PEL Shimura variety of type $\left( \text{A} \right)$ or $\left( \text{C} \right)$ such that $p$ is an unramified prime for the PEL datum and such that the ordinary locus is dense in the reduction of $X$. 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 parameterize finite slope systems of eigenvalues appearing in the space of families of cuspidal forms.

Keywords

11F5511F33p-adic modular formseigenvarietiesPEL-type Shimura varieties
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 6 , 01 December 2016 , pp. 1227 - 1256
Copyright
Copyright © Canadian Mathematical Society 2016

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