A non-trivial family of bundles fixed by the square of Frobenius

Abstract

One shows the existence of a smooth projective curve over $\text{F}{}_2$ and of representations of the arithmetic fundamental group of X⊗k with values in $\text{SL}{}_2\text{(k[[t]])}$, with k suitable finite field of characteristic 2, such that the image of the geometric fundamental group is infinite. This gives a negative answer to a question of A.J. de Jong.