Moduli of rank 1 isocrystals

TL;DR

This paper characterizes rank 1 isocrystals on proper curves within the de Rham moduli space, confirming Deligne's conjecture in this setting by comparing their $\, ext{l}$-adic cohomologies using Berkovich space theory.

Contribution

It expresses rank 1 isocrystals as a subset of the de Rham moduli space and verifies Deligne's conjecture through cohomological comparison.

Findings

Confirmed Deligne's conjecture for rank 1 isocrystals.

Established a link between isocrystals and de Rham moduli space.

Compared $\, ext{l}$-adic cohomologies using Berkovich spaces.

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.