The classical affine cubic surface of Markoff has a well-known interpretation as a moduli space for local systems on the once-punctured torus. We show that the analogous moduli spaces for general topological surfaces form a rich family of log Calabi-Yau varieties, where a structure theorem for their integral points can be established using mapping class group descent. Related analysis also yields new results on the arithmetic of algebraic curves in these moduli spaces, including finiteness of imaginary quadratic integral points for non-special curves.
We report on some recent work with Peter Sarnak. For integers $k$, we consider the affine cubic surfaces $V_k$ given by $M(x) = x_1^2 + x_2 + x_3^2 − x_1 x_2 x_3 = k$. Then for almost all $k$, the Hasse Principle holds, namely that $V_k(Z)$ is non-empty if $V_k(Z_p)$ is non-empty for all primes $p$. Moreover there are infinitely many $k$'s for which it fails. There is an action of a non-linear group on the integral points, producing finitely many orbits. For most $k$, we obtain an exact description of these orbits, the number of which we call "class numbers".
The Graduate Center, The City University of New York
December 8, 2017
Markoff triples are integer solutions of the equation $x^2+y^2+z^2 = 3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond.