On matrices commuting with their Frobenius

Abstract: The Frobenius of a matrix $M$ with coefficients in $\bar{\mathbb F}_p$ is the matrix $\sigma(M)$ obtained by raising each coefficient to the $p$-th power. We consider the question of counting matrices with coefficients in $\mathbb F_q$ which commute with their Frobenius, asymptotically when $q$ is a large power of $p$. We give answers for matrices of size $2$, for diagonalizable matrices, and for matrices whose eigenspaces are defined over $\mathbb F_p$. Moreover, we explain what is needed to solve the case of general matrices. We also solve (for both diagonalizable and general matrices) the corresponding problem when one counts matrices $M$ commuting with all the matrices $\sigma(M)$, $\sigma^2(M)$, $\ldots$ in their Frobenius orbit.