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Let $g$ be a random matrix distributed according to uniform probabilitymeasure on the finite general linear group $\mathrm{GL}_n(\mathbb{F}_q)$. Weshow 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 showthat nontrivial linear combinations of $\mathrm{Tr}(g^1),\ldots,\mathrm{Tr}(g^k)$ equidistribute as long as $\log k =o(n)$ and this range issharp as well. Previously equidistribution of either a single trace or a linearcombination of traces was only known for $k \le c_q n$, where $c_q$ depends on$q$, due to work of the first-named author and Rodgers.  We reduce the problem to exhibiting cancellation in certain short charactersums in function fields. For the equidistribution of $\mathrm{Tr}(g^k)$ we endup showing that certain $\textit{explicit}$ 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 theclassical range $\log k =o(n)$ due to Montgomery and Vaughan. To study thesesums we build on the argument of Montgomery and Vaughan but exploit additionalsymmetry present in the considered sums.

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