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p–adic Families of Cohomological Modular Forms for Indefinite Quaternion Algebras and the Jacquet–Langlands Correspondence
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- Journal:
- Canadian Journal of Mathematics / Volume 68 / Issue 5 / 01 October 2016
- Published online by Cambridge University Press:
- 20 November 2018, pp. 961-998
- Print publication:
- 01 October 2016
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- Article
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- You have access
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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.
