Let $X$ be a smooth projective curve over a finite field of characteristic $p$, with Hasse-Witt invariant $\lambda>1$.
For $\ell\neq p$, the Frobenius action on the etale cohomology group $H^1(X,\mathbb Q_\ell)$ is semisimple. Is the statement still true when $\ell=p$?
Does anyone have a good reference, or a counterexample? Thanks.