Counting matrix solutions of Diophantine equations

A central subject in number theory is counting solutions of polynomial equations in a finite field. In the talk, I will recall the famous Sato--Tate conjecture in this theme and several of its variants. Can matrices be involved in this story, and what extra things do they bring? To answer this question, I will define the notion of n x n matrix solutions, and present results about their counts, where both arithmetic (e.g., Sato--Tate) and combinatorics (e.g., partitions of n) play a role.

Part of this talk is based on joint work with Ono and Saad and joint work with Jiang.