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

Greenberg, Matthew ; Seveso, Marco

Canadian journal of mathematics, Tome 68 (2016) no. 5, pp. 961-998

Résumé

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.

Détail

DOI : 10.4153/CJM-2015-062-x

Mots-clés : 11F11, 11F67, 11F85, modular forms, p–adic families, Jacquet–Langlands correspondence, Shimura curves, eigencurves

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