Abstract: Recent work has greatly enhanced our understanding of the ${\mathbb F}_p$-points on the Markoff surface defined by $x^2 + y^2 + z^2 = 3xyz$.  In particular, we now know, thanks to work of Bourgain-Gamburd-Sarnak, that the action on a natural quotient is almost always transitive, and by Carmon-Meiri-Puder, that the action is almost always as large as possible.  The goal of this talk is to prove a result analogous to that of Carmon-Meiri-Puder for the K3 surfaces of the form $(E x E)/\pm 1$, where $E$ is an elliptic curve defined over $\mathbb Q$.  In order to do this we will prove a surprising property of points on certain rational curves on this surface.  This is joint work with Owen Patashnick.

Zoom:  https://arizona.zoom.us/j/83621467612