Anderson-Stark units for $\mathbb F_q[\theta]$

TL;DR

This paper explores the arithmetic properties of special values of new $L$-functions, introducing Anderson-Stark units that encode these values and relate $L$-functions to polylogarithms.

Contribution

It introduces Anderson-Stark units and demonstrates their role in expressing $L$-function values as sums of polylogarithms, advancing understanding of function field arithmetic.

Findings

Special values of new $L$-functions are encoded in Anderson-Stark units

$L$-functions can be expressed as sums of polylogarithms using these units

The work links $L$-values, Anderson-Stark units, and polylogarithms in function field arithmetic

Abstract

We investigate the arithmetic of special values of a new class of $L$-functions recently introduced by the second author. We prove that these special values are encoded in some particular polynomials which we call Anderson-Stark units. We then use these Anderson-Stark units to prove that $L$-functions can be expressed as sums of polylogarithms.