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?
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4$\begingroup$ If we restrict to line bundles, this is asking if the $\mathbb F_q$ points of the Jacobian of the curve can contain exactly one point. The Lang-Weil bounds rule this out for for large enough $q$ and the only case I would be worried about would be $q =2,3,5$ and a genus one elliptic curve. There are examples for $q=2,3$ as here: math.stackexchange.com/questions/3001457/… $\endgroup$– AsvinCommented Jan 8, 2021 at 5:52
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1$\begingroup$ @Asvin Can you explain your first sentence? The condition only says that there are no $p-1$ torsion points which are $\mathbb{F}_q$ rational other than the trivial line bundle. $\endgroup$– MohanCommented Jan 8, 2021 at 19:48
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$\begingroup$ @Mohan I might have missed something but to say that a line bundle $\mathscr L/X$ is fixed by Frobenius is to say exactly that as a point on the Pic(X), it is fixed by the Frobenius action and hence is an actual $\mathbb F_q$ point, no? Where is the $p-1$ torsion coming from? $\endgroup$– AsvinCommented Jan 8, 2021 at 21:02
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2$\begingroup$ @Asvin The morphism induced on Pic(X) by Frobenius of X is different from the internal Frobenius of Pic(X): for a test scheme $S$ the set $Pic(X)(S)$ is the set of isomorphisms classes of line bundles on $X\times S$ modulo those coming from $S$ and F corresponds to pullback of line bundles along $F_X\times 1$ while the internal Frobenius of Pic(X) corresponds to pulling back along $1\times F_S$. Explicitly, as Mohan says, for a line bundle $L$ the pullback $F^*L$ is canonically isomorphic to $L^{\otimes p}$ (e.g. because the gluing invertible functions are getting raised to p-th power by F). $\endgroup$– SashaPCommented Jan 8, 2021 at 22:26
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$\begingroup$ @SashaP Thank you so much! $\endgroup$– AsvinCommented Jan 8, 2021 at 22:50
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1 Answer
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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).
