On matrices commuting with their Frobenius

The Frobenius of a matrix M with coefficients in F ‾ p is the matrix σ ( M ) obtained by raising each coefficient to the p -th power. We consider the question of counting matrices with coefficients in 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 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 σ ( M ) , σ 2 ( M ) , … in their Frobenius orbit.