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

  • Matthew Greenberg (a1) and Marco Seveso (a2)

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.

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