Mellin transform formulas for Drinfeld modules

TL;DR

This paper develops explicit formulas for the logarithms of Drinfeld modules using motivic maps and rigid analytic trivializations, linking special values of Goss L-functions to Drinfeld periods and extending to tensor powers of the Carlitz module.

Contribution

It introduces a new framework for expressing Drinfeld module logarithms and special L-values via motivic evaluations, providing characteristic-p analogues of classical zeta function formulas.

Findings

Formulas for Drinfeld module logarithms using motivic maps

Expression of Goss L-values as Drinfeld periods and trivializations

Application to tensor powers of the Carlitz module as Tate twists

Abstract

We introduce formulas for the logarithms of Drinfeld modules using a framework recently developed by the second author. We write the logarithm function as the evaluation under a motivic map of a product of rigid analytic trivializations of $t$-motives. We then specialize our formulas to express special values of Goss $L$-functions as Drinfeld periods multiplied by rigid analytic trivializations evaluated under this motivic map. We view these formulas as characteristic-$p$ analogues of integral representations of Hasse-Weil type zeta functions. We also apply this machinery for Drinfeld modules tensored with the tensor powers of the Carlitz module, which serves as the Tate twist of a Drinfeld module.