Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

Abstract

Let $g$ be a random matrix distributed according to uniform probability measure on the finite general linear group ${\mathrm{GL}}_n({\mathbb{F}}_q)$. We show that $\mathrm{Tr}(g^k)$ equidistributes on ${\mathbb{F}}_q$ as $n\to \infty$ as long as $\log k=o(n^2)$ and that this range is sharp. We also show that nontrivial linear combinations of $\mathrm{Tr}(g^1),\dots ,\mathrm{Tr}(g^k)$ equidistribute as long as $\log k=o(n)$ and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for $k\le c_{q}n$, where $c_q$ depends on $q$, due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of $\mathrm{Tr}(g^k)$ we end up showing that certain explicit character sums modulo $T^{k+1}$ exhibit cancellation when averaged over monic polynomials of degree $n$ in ${\mathbb{F}}_q[T]$ as long as $\log k=o(n^2)$. This goes far beyond the classical range $\log k=o(n)$ due to Montgomery and Vaughan. To study these sums we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.

19 pages, accepted version. Includes 1-page appendix on character sums not included in published version