# Differential Characters of Drinfeld Modules and de Rham Cohomology

## TL;DR

This paper introduces differential characters for Drinfeld modules, establishing their structure, revealing differential modular functions, and constructing an associated $F$-crystal linked to de Rham cohomology, with implications for function field arithmetic.

## Contribution

It defines differential characters for Drinfeld modules, analyzes their structure, and constructs an $F$-crystal connecting to de Rham cohomology, extending analogies from elliptic curves.

## Key findings

- Existence of a family of differential modular functions
- Structure of the group of differential characters determined
- Construction of a canonical $F$-crystal related to de Rham cohomology

## Abstract

We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical $F$-crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This $F$-crystal is of a differential-algebraic nature, and the relation to the classical cohomological realizations is presently not clear.