Explicit formulas for Drinfeld modules and their periods

TL;DR

This paper derives explicit series formulas for exponential and logarithm functions of Drinfeld modules, enabling computation of periods and criteria for supersingularity, thus advancing understanding of their structure.

Contribution

It provides new explicit formulas for Drinfeld module functions and a method to compute periods and identify supersingularity, generalizing known results for the Carlitz module.

Findings

Explicit series expansions for Drinfeld module functions

A procedure for computing rank 2 Drinfeld module periods

A criterion for supersingularity in Drinfeld modules

Abstract

We provide explicit series expansions for the exponential and logarithm functions attached to a rank r Drinfeld module that generalize well known formulas for the Carlitz exponential and logarithm. Using these results we obtain a procedure and an analytic expression for computing the periods of rank 2 Drinfeld modules and also a criterion for supersingularity.