The spectral $p$-adic Jacquet-Langlands correspondence and a question of Serre

TL;DR

This paper proves an isomorphism between the completed Hecke algebras of $p$-adic modular forms and quaternionic automorphic forms, answering a question by Serre and providing new insights into Galois representations.

Contribution

It establishes a geometric proof of the isomorphism, independent of classical Jacquet-Langlands, linking $p$-adic modular forms and quaternionic automorphic forms.

Findings

Isomorphism between completed Hecke algebras established

New geometric proof of Galois representations attached to quaternionic eigenforms

Answer to Serre's 1987 question on $p$-adic automorphic forms

Abstract

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.