Date：June 19th，16:00-17:00

Location：Lecture hall210,  Building 2, Hainayuan

Abstract：

The Markoff equation x^2 + y^2 + z^2 - 3xyz = 0 first arose in Markoff's investigations into Diophantine approximation and quadratic forms, but has since appeared in a variety of contexts, ranging from solutions to Painleve VI differential equations to the derived categories of varieties. In 2016 Bourgain, Gamburd, and Sarnak studied the Diophantine properties of the Markoff equation, and conjectured that its integral points surject onto its mod p points for every prime (i.e., it satisfies strong approximation). Moreover, they prove the conjecture for all but a sparse (but possibly infinite) set of primes p. In this talk we will show how to prove the conjecture for all but finitely many primes p. The proof involves relating the conjecture to the connectivity of certain Hurwitz moduli spaces of SL(2,p)-covers of elliptic curves, and proving a divisibility theorem on the degree of the forgetful map from the Hurwitz space to the moduli space of elliptic curves. This divisibility theorem provides sufficient rigidity to bootstrap Bourgain, Gamburd and Sarnak's results to a solution of the conjecture for all but finitely many primes, and more generally hints at a fruitful connection between Hurwitz spaces and the Diophantine geometry of character varieties.

Introduction of the Speaker：

Will is a Hill assistant professor at Rutgers University in New Jersey. He is interested in arithmetic geometry, especially the interplay between Hurwitz spaces, elliptic curves, and noncongruence modular curves.

He was an NSF postdoctoral fellow from 2018-2021, and a member at the IAS in Princeton from 2016-2018 and 2021-2022. Recently, his paper Nonabelian level structures, Nielsen equivalence, and Markoff triples was accepted to the Annals of Math.

Welcome！