Differential characters of Drinfeld modules and de Rham cohomology

Borger, James; Saha, Arnab

doi:10.2140/ant.2019.13.797

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Differential characters of Drinfeld modules and de Rham cohomology

James Borger and Arnab Saha

Vol. 13 (2019), No. 4, 797–837

DOI: 10.2140/ant.2019.13.797

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.

Keywords

arithmetic geometry, number theory, algebraic geometry, arithmetic jet spaces, Witt vectors, Drinfeld modules, differential characters, de Rham cohomology

Mathematical Subject Classification 2010

Primary: 11G99

Secondary: 14L05

Milestones

Received: 3 September 2017

Revised: 26 November 2018

Accepted: 22 February 2019

Published: 6 May 2019

Authors

James Borger
Mathematical Sciences Institute Australian National University Canberra ACT Australia

Arnab Saha
Max Planck Institute for Mathematics Bonn Germany

Vol. 13, No. 4, 2019