P–adic Families of Cohomological Modular Forms for Indefinite Quaternion Algebras and the Jacquet–Langlands Correspondence. (1st October 2016) - Greenberg, Matthew; Seveso, Marco
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.