Vol. 13, No. 4, 2019

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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
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.

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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