Special values of Goss $L$-series attached to Drinfeld modules of rank 2

TL;DR

This paper provides explicit formulas for special values of Goss $L$-series associated with rank 2 Drinfeld modules, linking them to polylogarithms and logarithm coefficients, inspired by classical number theory.

Contribution

It introduces explicit formulas for Goss $L$-series values at positive integers for rank 2 Drinfeld modules, under specific coefficient conditions.

Findings

Explicit formulas for $L$-series values at positive integers.

Connection between $L$-series values, polylogarithms, and logarithm coefficients.

Results applicable when $2n+1 \

Abstract

Inspired by the classical setting, Goss defined $L$-series attached to Drinfeld modules. In this paper, for a fixed choice of a power $q$ of a prime number and a given Drinfeld module $\phi$ of rank 2 with a certain condition on its coefficients, we give explicit formulas for the values of Goss $L$-series attached to $\phi$ at positive integers $n$ such that $2n+1\leq q$ in terms of polylogarithms and coefficients of the logarithm series of $\phi$.