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 ${}E_{1}\subset \ldots \subset E_{t}$ be a strong Harder-Narasimhan filtration of ${}F^{e*}(E)$. We choose ${}i$ such that ${}E_{i}/E_{i-1}$ has degree ${}\geq 0$ and that ${}E_{i+1}/E_{i}$ has degree ${}<0$. We set ${}{\mathcal {Q}}=F^{e*}(E)/E_{i}$.

Then the following are equivalent.

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