##### Department of Mathematics,

University of California San Diego

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### Math 211 - Algebra Seminar

## Matt Litman

#### UC Davis

## 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.

### January 10, 2022

### 2:00 PM

Zoom meeting:

ID 939 5383 2894

Password: structures

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