Endlicher Körper/Projektive glatte Kurve/Vektorbündel/Endliche Annulation/Harder-Narasimhan-Kriterium/Fakt/en

Let $K$ denote a finite field (or the algebraic closure of a finite field) and let $X$ be a smooth projective curve over $K$. Let $E$ be a locally free sheaf over $X$ and let  $c\in H^1(X,E)$  denote a cohomology class. Let  $E_1\subset \dots \subset E_t$  be a strong Harder-Narasimhan filtration of $F^{e\ast }(E)$. We choose $i$ such that $E_i/E_{i-1}$ has degree $\ge 0$ and that $E_{i+1}/E_i$ has degree $<0$. We set  $𝒬=F^{e\ast }(E)/E_i$. 

Then the following are equivalent.

1. The class $c$ can be annihilated by a finite morphism.
2. Some Frobenius power of the image of $F^{e\ast }(c)$ inside $H^1(X,𝒬)$ is $0$.