Diophantine analysis and Arthur's trace formula
Yuchan Lee
TL;DR
The paper develops a Diophantine-analytic framework to count integral points on a $G$-homogeneous space $X\cong G_\gamma\backslash G$, with $G$ simply connected and semisimple, and a centralizer $G_\gamma$ that is an anisotropic maximal torus. It expresses the leading asymptotics of the height-bounded count $N(X;f_{\mathbf{X},T})$ in terms of $\kappa$-orbital integrals, connecting the count to the stabilized geometric side of Arthur’s trace formula. Through endoscopic transfer and the fundamental lemma, the $\kappa$-orbital contributions are rewritten as stable orbital integrals on endoscopic groups, highlighting a deep link between Diophantine counting and automorphic data. The framework is then specialized to the SL_n-case of matrices with a fixed irreducible characteristic polynomial $\chi$, yielding explicit asymptotics that generalize Eskin–Mozes–Shah by relaxing key real/arithmetical constraints and expressing constants via local GL_n orbital integrals and standard invariants (discriminants, regulators, zeta-values).
Abstract
Let $X$ be a $G$-homogeneous space over a number field $k$ such that $X\cong G_γ\backslash G$. Here, $G$ is a simply connected semisimple group over $k$ and $γ\in G(k)$ whose centralizer $G_γ$ 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 $κ$-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 $γ$ 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 $χ(x)$ of degree $n$. This result generalizes a case studied by Eskin-Mozes-Shah (1996).