Vector bundles that are fixed under pull-back by the absolute Frobenius Are there algebraic projective curves over finite fields other than $\mathbb{P^1}$ that if a vector bundle on it, is stable under Frobenius i.e. $F^*E\cong E$ implies that $E$ is a trivial bundle? If so, does every algebraic curve admit an etale cover of this form?
 A: For a finite flat cover $\pi:Y\to X$ the pushforward $E:=\pi_*\mathcal{O}_Y$ comes with a morphism $F^*E\to E$ induced by the Frobenius on $Y$. If $\pi$ is etale this morphism is an isomorphism: over affine charts $X=Spec\, A$, $Y=Spec\, B$ we want to show that the map $B\otimes_{A,F_A}A\xrightarrow{b\otimes a\mapsto b^pa} B$ is an isomorphism. This is true as this is a map between finite etale $A$-algebras of the same degree.
Let $X$ be a smooth projective geometrically connected curve over $K$ of genus $g>0$. If $\pi:Y\to X$ is a finite etale morphism that does not have the form $X_L\to X$ (the base change of $Spec\, L\to Spec\, K$ to $X$) for a finite extension $L\supset K$ then $E=\pi_*\mathcal{O}_Y$ is necessarily non-trivial. Indeed, if $\pi$ decomposes as $Y\to X_{L}\to X$ where $L\supset K$ is a finite extension of degree $d_1$ and $Y$ is a geometrically irreducible curve over $L$ with $Y\to X_L$ of degree $d_2$ then, by Riemann-Hurwitz, $\dim_L H^1(Y, \mathcal{O}_Y)=d_2(g-1)+1$ so $\dim_KH^1(X,\pi_*\mathcal{O}_Y)=d_1(d_2(g-1)+1)$ while a trivial bundle of rank $d_1d_2$ on $X$ has first cohomology of dimension $d_1d_2g$ so $\pi_*\mathcal{O}_Y$ can't be trivial as long as $d_2>0$.
Any curve of genus $>0$ admits an etale cover which is not just an extension of the base field, so $\mathbb{P}^1$ is the only smooth projective curve over a finite field with that property (over characteristic $p$ fields with non-trivial $2$-torsion in Brauer group forms of $\mathbb{P}^1$ would also give examples).
