Joint IAS/Princeton University Number Theory Seminar

Following Bourgain, Gamburd, and Sarnak, we say that the Markoff equation $x^2 + y^2 + z^2 - 3xyz = 0$ satisfies strong approximation at a prime $p$ if its integral points surject onto its $F_p$ points. In 2016, Bourgain, Gamburd, and Sarnak were able to establish strong approximation at all but a sparse (but infinite) set of primes, and conjecture that it holds at all primes. Building on their results, in this talk I will explain how to obtain strong approximation for all but a finite and effectively computable set of primes, thus reducing the conjecture to a finite computation. The key result amounts to establishing a congruence on the degree of a certain line bundle on the moduli stack of elliptic curves with $SL(2,p)$-structures. To make contact with the Markoff equation, we use the fact that the Markoff surface is a level set of the character variety for $SL(2)$ representations of the fundamental group of a punctured torus, and that the strong approximation conjecture can be expressed in terms of the mapping class group action on the character variety, which in turn also determines the geometry of the moduli stack of elliptic curves with $SL(2,p)$-structures. As time allows we will also describe a number of applications.