Title: 

Density of integral points on character varieties

Abstract: 

Given a smooth complex algebraic variety Y and an algebraic group G, a (relative) G-character variety is a moduli space of G-local systems with specified behavior at the boundary of a compactification of Y. A well-known example of SL_2-character varieties are Markoff type cubic surfaces, and in recent years the study of their integral points has attracted much attention, starting with the work of Bourgain, Gamburd and Sarnak. In my talk I will present joint work with Daniel Litt where we prove that, for any algebraic variety Y, integral points are potentially Zariski dense in the relative character varieties parametrizing SL_2-local systems on Y with fixed algebraic integer traces along the boundary components. The proof first treats the case of Y a Riemann surface, where we construct one initial integral point and then produce a Zariski dense set of them by exploiting the dynamics of the mapping class group action on the character variety. The case of general Y is reduced to the Riemann surface case by using work of Corlette and Simpson.

https://www.unine.ch/math/seminaire-algebre-et-geometrie-de-neuchatel/