On the transcendence of special values of Goss $L$-functions attached to Drinfeld modules

TL;DR

This paper proves the transcendence of special values of Goss $L$-functions associated with Drinfeld modules over function fields, extending the understanding of their algebraic independence and transcendental nature.

Contribution

It establishes the transcendence of Goss $L$-function values at positive integers for certain Drinfeld modules, a significant advancement in function field arithmetic.

Findings

Transcendence of Goss $L$-values at positive integers for Drinfeld modules of rank ≥ 2.

Proof that these values are transcendental over the base function field.

Extension of transcendence results to exterior powers of associated $t$-motives.

Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the rational function field $K:=\mathbb{F}_q(\theta)$. For a Drinfeld module $\phi$ defined over $K$, we study the transcendence of special values of the Goss $L$-function attached to the abelian $t$-motive $M_{\phi}$ of $\phi$. Moreover, when $\phi$ is a Drinfeld module of rank $r\geq 2$ defined over $K$ which has everywhere good reduction, we prove that the value of the Goss $L$-function attached to the $(r-1)$-st exterior power of $M_{\phi}$ at any positive integer is transcendental over $K$.