Joint distribution of inverses in matrix groups over finite fields

Abstract

We study the joint distribution of the solutions to the equation $gh=x$ in $G(\mathbb {F}_p)$ as $p\rightarrow \infty$, for any fixed $x\in G(\mathbb {Z})$, where $G={\text {GL}}_n$, ${\text {SL}}_n$, ${\text {Sp}}_{2n}$ or ${\text {SO}}_{n}^\pm$. In the special linear case, this answers in particular a question raised by Hu and 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 $g\in {\text {GL}}_n(\mathbb {F}_p)$ 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.