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p–adic Families of Cohomological Modular Forms for Indefinite Quaternion Algebras and the Jacquet–LanglandsCorrespondence

Published online by Cambridge University Press:  20 November 2018

Matthew Greenberg  and
Marco Seveso
Matthew Greenberg
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4 e-mail: mgreenbe@ucalgary.ca
Marco Seveso
Affiliation:
Dipartimento di Matematica, Universitá di Milano, 20133 Milano, Italia e-mail: seveso.marco@gmail.com
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Abstract

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We use the method of Ash and Stevens to prove the existence of small slope $p$-adic families of cohomological modular forms for an indefinite quaternion algebra $B$. We prove that the Jacquet–Langlands correspondence relating modular forms on $\text{G}{{\text{L}}_{\text{2}}}/\mathbb{Q}$ and cohomomological modular forms for $B$ is compatible with the formation of $p$-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.

Keywords

11F1111F6711F85modular formsp–adic familiesJacquet–Langlands correspondenceShimura curveseigencurves
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 5 , 01 October 2016 , pp. 961 - 998
Copyright
Copyright © Canadian Mathematical Society 2016

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