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DTSTAMP:20180205T081348Z
DTSTART;TZID=America/Los_Angeles:20180212T151000
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SUMMARY:Arithmetic Geometry and Number Theory RTG Seminar: The $p$-adic Jacquet-Langlands correspondence and a question of Serre
UID:115327-ucb-events-calendar@berkeley.edu
ORGANIZER;CN="UC Berkeley Calendar Network":
LOCATION:748 Evans Hall
DESCRIPTION:Sean Howe\, Stanford University\n\nSeminar 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.\n\nAbstract 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.\n\nAbstract 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.
URL:http://events.berkeley.edu/index.php/calendar/sn/pubaff.html?event_ID=115327&view=preview
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