Joint distribution of inverses in matrix groups over finite fields

TL;DR

This paper investigates the distribution of solutions to the equation gh=x in matrix groups over finite fields as the prime p grows large, using advanced algebraic geometry techniques to improve error estimates and establish new existence results.

Contribution

It provides new asymptotic distribution results for solutions in matrix groups over finite fields, answering open questions and improving error bounds with geometric methods.

Findings

Distribution of solutions approaches uniformity as p increases

Existence of matrices with entries and inverses bounded in specified intervals

Improved error terms over previous results in special linear groups

Abstract

We study the joint distribution of the solutions to the equation $gh=x$ in $G(\mathbb{F}_p)$ as $p\to\infty$, for any fixed $x\in G(\mathbb{Z})$, where $G=\operatorname{GL}_n$, $\operatorname{SL}_n$, $\operatorname{Sp}_{2n}$ or $\operatorname{SO}_{n}^\pm$. In the special linear case, this answers in particular a question raised by S. Hu and Y. Li, and improves their error terms. Similar results are derived in certain subgroups, and when the entries of $g,h$ lie in fixed intervals. The latter shows for example the existence of $g\in\operatorname{GL}_n(\mathbb{F}_p)$ such that $g,g^{-1}$ have all entries in $[0, c_np^{1-1/(2n^2+2)+\varepsilon}]$ for some absolute constant $c_n>0$. The key for these results is to use Deligne's extension of the Weil conjectures on a sheaf on $G$, along with the stratification theorem of Fouvry, Katz and Laumon, instead of reducing to bounds on classical Kloosterman sums.