Deninger et Werner ont développé un analogue pour les courbes $p$-adiques de la correspondance classique de Narasimhan et Seshadri entre les fibrés vectoriels stables de degré $0$ et les représentations unitaires du groupe fondamental topologique pour une courbe complexe propre et lisse. Par transport parallèle, ils ont associé fonctoriellement à chaque fibré vectoriel sur une courbe $p$-adique, dont la réduction est fortement semi-stable de degré $0$, une représentation $p$-adique du groupe fondamental de la courbe. Ils se sont posé quelques questions : leur foncteur est-il pleinement fidèle ? La cohomologie des systèmes locaux fournis par celui-ci admet-elle une filtration de Hodge-Tate ? Leur construction est-elle compatible avec la correspondance de Simpson $p$-adique développée par Faltings ? Nous répondons à ces questions dans cet article.

We prove that the direct image complex for the $D$-twisted $\text{SL}_{n}$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $\text{GL}_{n}$ is due to Chaudouard and Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\unicode[STIX]{x1D6FF}$-regularity results for some auxiliary weak abelian fibrations.

Let X be a smooth projective curve of genus $g > 1$ over an algebraically closed field of positive characteristic. This paper is a study of a natural stratification, defined by the absolute Frobenius morphism of X, on the moduli space of vector bundles. In characteristic two, there is a complete classification of semi-stable bundles of rank 2 which are destabilized by Frobenius pull-back. We also show that these strata are irreducible and obtain their respective dimensions. In particular, the dimension of the locus of bundles of rank two which are destabilized by Frobenius is $3g-4$. These Frobenius destabilized bundles also exist in characteristics two, three and five with ranks 4, 3 and 5, respectively. Finally, there is a connection between (pre)-opers and Frobenius destabilized bundles. This allows an interpretation of some of the above results in terms of pre-opers and provides a mechanism for constructing Frobenius destabilized bundles in large characteristics.

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.

The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah and Bott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Künneth components of the Chern classes of the universal bundle.

This paper studies the larger, non-compact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle $K$. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes.

The spaces of Higgs bundles with values in $K(n)$ for $n > 0$ turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes.