NT Seminar this week

[Jayadev Athreya]

Matt Litman, UC Davis

Tuesday, November 30⋅11:00 – 11:50
Matt Litman, Davis

Title - Markoff-type K3 Surfaces: Local and Global Finite Orbits

Abstract - Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran.

Zoom details: Jayadev S Athreya is inviting you to a scheduled Zoom meeting.

Topic: UW Number Theory Seminar (Litman)
Time: Nov 30, 2021 11:00 AM Pacific Time (US and Canada)

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Meeting ID: 987 7666 5215

Jayadev Athreya

Professor of Mathematics

Professor of the Comparative History of Ideas

University of Washington

Interim Director, Pacific Institute for the Mathematical Sciences

[Jayadev Athreya]