Moduli of rank 1 isocrystals

Abstract

In this article we express the set of rank 1 isocrystals on a proper curve as a subset of the de Rham moduli space, defined by Simpson and Langer. Using results from the theory of Berkovich spaces, we compare the $\ell$-adic cohomology of this subspace with the $\ell$-adic cohomology of the whole moduli space. This confirms a conjecture of Deligne in the rank 1 case and explains one of his examples in this case from the point of view of isocrystals.