Diophantine analysis and Arthur's trace formula

Abstract

Let $X$ be a $G$-homogeneous space over a number field $k$ such that $X\cong G_\gamma\backslash G$. Here, $G$ is a simply connected semisimple group over $k$ and $\gamma\in G(k)$ whose centralizer $G_\gamma$ is a maximal torus in $G$ which is anisotropic over $k$. We formulate the asymptotic for the number of integral points on $X$ bounded by a fixed norm $T>0$ as $T\rightarrow \infty$ in terms of $\kappa$-orbital integrals, which play a role in the stabilization of Arthur's trace formula. This formula coincides with the contribution of the stable conjugacy class of $\gamma$ to the geometric side of the trace formula. As an application, we obtain an asymptotic formula for the number of $n \times n$ matrices over the ring of integers $\mathcal{O}_k$ whose characteristic polynomial equals a fixed irreducible polynomial $\chi(x)$ of degree $n$. This result generalizes a case studied by Eskin-Mozes-Shah (1996).