Title: Diophantine analysis and Arthur's Trace formula

Abstract:

A Diophantine problem concerning the asymptotic behavior of integral points of a variety $X$ has been extensively studied over $\mathbb{Q}$, and further extended to general number fields when $X$ is a homogeneous space of a simply connected semisimple group. 

On the other hand, Arthur's trace formula, a central tool in the Langlands program, provides a powerful framework for studying automorphic representations.

In this talk, I will present my recent work connecting these two themes: Diophantine analysis and the geometric side of Arthur’s trace formula. 

As an application, I will describe an asymptotic formula for the number of integral matrices with a fixed characteristic polynomial. 

If time permits, I will also discuss the underlying heuristic from the perspective of the $L^2$-expansion.