We deal first with the situation of a strongly semistable sheaf ${\displaystyle {}{\mathcal {S}}}$ of degree ${\displaystyle {}0}$. The following two results are due to Lange and Stuhler. We say that a locally free sheaf is étale trivializable if there exists a finite étale morphism ${\displaystyle {}\varphi \colon C'\rightarrow C}$ such that ${\displaystyle {}\varphi ^{*}({\mathcal {S}})\cong {\mathcal {O}}_{C'}^{r}}$. Such bundles are directly related to linear representations of the étale fundamental group.

## Lemma

Let ${\displaystyle {}K}$ denote a finite field (or the algebraic closure of a finite field) and let ${\displaystyle {}X}$ be a smooth projective curve over ${\displaystyle {}K}$. Let ${\displaystyle {}{\mathcal {S}}}$ be a locally free sheaf over ${\displaystyle {}X}$. Then ${\displaystyle {}{\mathcal {S}}}$ is étale trivializable if and only if there exists some ${\displaystyle {}n}$ such that ${\displaystyle {}F^{n*}{\mathcal {S}}\cong {\mathcal {S}}}$.

## Theorem

Let ${\displaystyle {}K}$ denote a finite field (or the algebraic closure of a finite field) and let ${\displaystyle {}X}$ be a smooth projective curve over ${\displaystyle {}K}$. Let ${\displaystyle {}{\mathcal {S}}}$ be a strongly semistable locally free sheaf over ${\displaystyle {}X}$ of degree ${\displaystyle {}0}$. Then there exists a finite morphism

${\displaystyle \varphi \colon Y\longrightarrow X}$

such that ${\displaystyle {}\varphi ^{*}({\mathcal {S}})}$ is trivial.

### Proof

We consider the family of locally free sheaves ${\displaystyle {}F^{e*}({\mathcal {S}})}$, ${\displaystyle {}e\in \mathbb {N} }$. Because these are all semistable of degree ${\displaystyle {}0}$, and defined over the same finite field, we must have (by the existence of the moduli space for vector bundles) a repetition, i.e.

${\displaystyle {}F^{e*}({\mathcal {S}})\cong F^{e'*}({\mathcal {S}})\,}$

for some ${\displaystyle {}e'>e}$. By fact, the bundle ${\displaystyle {}F^{e*}({\mathcal {S}})}$ admits an étale trivialization ${\displaystyle {}\varphi \colon Y\rightarrow X}$. Hence the finite map ${\displaystyle {}F^{e}\circ \varphi }$ trivializes the bundle.

${\displaystyle \Box }$

## Theorem

Let ${\displaystyle {}K}$ denote a finite field (or the algebraic closure of a finite field) and let ${\displaystyle {}X}$ be a smooth projective curve over ${\displaystyle {}K}$. Let ${\displaystyle {}{\mathcal {S}}}$ be a strongly semistable locally free sheaf over ${\displaystyle {}X}$ of nonnegative degree and let ${\displaystyle {}c\in H^{1}(X,{\mathcal {S}})}$ denote a cohomology class. Then there exists a finite morphism

${\displaystyle \varphi \colon Y\longrightarrow X}$

such that ${\displaystyle {}\varphi ^{*}(c)}$ is trivial.

### Proof

If the degree of ${\displaystyle {}{\mathcal {S}}}$ is positive, then a Frobenius pull-back ${\displaystyle {}F^{e*}({\mathcal {S}})}$ has arbitrary large degree and is still semistable. By Serre duality we get that ${\displaystyle {}H^{1}(X,F^{e*}({\mathcal {S}}))=0}$. So in this case we can annihilate the class by an iteration of the Frobenius alone.

So suppose that the degree is ${\displaystyle {}0}$. Then there exists by fact a finite morphism which trivializes the bundle. So we may assume that ${\displaystyle {}{\mathcal {S}}\cong {\mathcal {O}}_{X}^{r}}$. Then the cohomology class has several components ${\displaystyle {}c_{i}\in H^{1}(X,{\mathcal {O}}_{X})}$ and it is enough to annihilate them separately by finite morphisms. But this is possible by the parameter theorem of K. Smith (or directly using Frobenius and Artin-Schreier extensions).

${\displaystyle \Box }$