Density of vector bundles periodic under the action of Frobenius

Abstract

Let X be a proper and smooth curve of genus g≥2 over an algebraically closed field k of positive characteristic. If k=F_p, it follows from Hrushovski's work on the geometry of difference schemes that the set of rank r vector bundles with trivial determinant over X that are periodic under the action of Frobenius is dense in the corresponding moduli space. Using the equivalence between Frobenius periodicity of a stable vector bundle and its triviality after pull-back by some finite etale cover of X (due to Lange and Stuhler) on the one hand, and specialization of the fundamental group on the other hand, we prove that the same result holds for any algebraically closed field of positive characteristic.