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

Abstract

AbstractWe use the method of Ash and Stevens to prove the existence of small slopep-adic families of cohomological modular forms for an indefinite quaternion algebraB. We prove that the Jacquet–Langlands correspondence relating modular forms onand cohomomological modular forms forBis compatible with the formation ofp-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.