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?



Vector bundles that are fixed under pull-back by the absolute Frobenius
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).


