A Motivic Pairing and the Mellin Transform in Function Fields

TL;DR

This paper introduces a new pairing related to Anderson A-modules, connecting exponential and logarithm functions with Mellin transform analogues in function fields, and applies these to zeta values.

Contribution

It defines a motivic pairing that links A-motives and their duals, leading to explicit formulas and Mellin transform analogues for zeta functions in function fields.

Findings

Derived explicit formulas for exponential and logarithm functions of Anderson A-modules.

Established a Mellin transform analogue for Carlitz zeta values.

Applied the pairing to analyze Carlitz multiple zeta values.

Abstract

We define two pairings relating the A-motive with the dual A-motive of an abelian Anderson A-module. We show that specializations of these pairings give the exponential and logarithm functions of this Anderson A-module, and we use these specializations to give precise formulas for the coefficients of the exponential and logarithm functions. We then use this pairing to express the exponential and logarithm functions as evaluations of certain infinite products. As an application of these ideas, we prove an analogue of the Mellin tranform formula for the Riemann zeta function in the case of Carlitz zeta values. We also give an example showing how our results apply to Carlitz multiple zeta values.