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

Let g$g$ be a random matrix distributed according to uniform probability measure on the finite general linear group GLn(Fq)$\mathrm{GL}_n(\mathbb {F}_q)$. We show that Tr(gk)$\mathrm{Tr}(g^k)$ equidistributes on Fq$\mathbb {F}_q$ as n→∞$n \rightarrow \infty$ as long as logk=o(n2)$\log k=o(n^2)$ and that this range is sharp. We also show that nontrivial linear combinations of Tr(g1),...,Tr(gk)$\mathrm{Tr}(g^1),\ldots, \mathrm{Tr}(g^k)$ equidistribute as long as logk=o(n)$\log k =o(n)$ 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⩽cqn$k \leqslant c_q n$, where cq$c_q$ depends on q$q$, 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 Tr(gk)$\mathrm{Tr}(g^k)$, we end up showing that certain explicit character sums modulo Tk+1$T^{k+1}$ exhibit cancellation when averaged over monic polynomials of degree n$n$ in Fq[T]$\mathbb {F}_q[T]$ as long as logk=o(n2)$\log k = o(n^2)$. This goes far beyond the classical range logk=o(n)$\log k =o(n)$ 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. [ABSTRACT FROM AUTHOR]