P–adic Families of Cohomological Modular Forms for Indefinite Quaternion Algebras and the Jacquet–Langlands Correspondence. (1st October 2016)

• Record Type: Journal Article
• Authors: Greenberg, Matthew; Seveso, Marco
• Is Part Of: Canadian journal of mathematics. Volume 68:Number 5(2016)
• Page Start: 961
• Page End: 998
• Publication Date: 2016-10-01
• Subjects: 11F11 -- 11F67 -- 11F85; modular forms -- p–adic families -- Jacquet–Langlands correspondence -- Shimura curves -- eigencurves; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2015-062-x
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: 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.