Vector bundles with a Frobenius structure on the punctured unit disc

Abstract

Let $\mathbb{C}$ be a complete non-archimedean-valued algebraically closed field of characteristic p > 0 and consider the punctured unit disc $\dot{D} \subset \mathbb{C}$. Let q be a power of p and consider the arithmetic Frobenius automorphism $\sigma_{\dot{D}}: x \mapsto x^{q^{-1}}$. A $\sigma$-bundle is a vector bundle $\mathcal{F}$ on $\dot{D}$ together with an isomorphism $\tau_\mathcal{F}: \sigma^\ast_{\dot{D}} \mathcal{F} \stackrel{\sim}{\longrightarrow} \mathcal{F}$. The aim of this article is to develop the basic theory of these objects and to classify them. It is shown that every $\sigma$-bundle is isomorphic to a direct sum of indecomposable $\sigma$-bundles $\mathcal{F}_{d,r}$ which depend only on rational numbers d/r. This result has close analogies with the classification of rational Dieudonné modules and of vector bundles on the projective line or on an elliptic curve. It has interesting consequences concerning the uniformizability of Anderson's t-motives that will be treated in a future paper.