Arithmetic Geometry and Number Theory RTG Seminar: The $p$-adic Jacquet-Langlands correspondence and a question of Serre

Seminar | February 12 | 3:10-5 p.m. | 748 Evans Hall

 Sean Howe, Stanford University

 Department of Mathematics

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.

Abstract of pretalk: We will recall the statement of the classical Jacquet-Langlands correspondence for automorphic forms on $GL_2$, and give some examples with modular forms. We will then recall some aspects of the mod $p$ geometry of modular curves and give a brief summary of Serre's mod $p$ Jacquet-Langlands correspondence.

Abstract of advanced talk: In a 1987 letter to Tate, Serre showed that the Hecke eigensystems appearing in mod p modular forms are the same as those appearing in mod p functions on a finite double coset constructed from the quaternion algebra ramified at p and infinity. At the end of the letter, he asked whether there might be a similar relation between $p$-adic modular forms and $p$-adic functions on the quaternion algebra. We show the answer is yes: the completed Hecke algebra of $p$-adic modular forms is the same as the completed Hecke algebra of naive $p$-adic automorphic functions on the quaternion algebra. The resulting $p$-adic Jacquet-Langlands correspondence is richer than the classical Jacquet-Langlands correspondence – for example, Ramanujan's delta function, which is invisible to the classical correspondence, appears. The proof is a lifting of Serre's geometric argument from characteristic $p$ to characteristic zero; the quaternionic double coset is realized as a fiber of the Hodge-Tate period map, and eigensystems are extended off of the fiber using Scholze's fake Hasse invariants.