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The spectral p-adic Jacquet–Langlands correspondence and a question of Serre
- Part of
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- Journal:
- Compositio Mathematica / Volume 158 / Issue 2 / February 2022
- Published online by Cambridge University Press:
- 11 April 2022, pp. 245-286
- Print publication:
- February 2022
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- Article
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We show that the completed Hecke algebra of
$p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous 
$p$-adic automorphic forms for the units of the quaternion algebra ramified at 
$p$ and 
$\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod 
$p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and 
$p$-adic quaternionic eigenforms.
