On the transcendence of twisted special $L$-values at non-negative integers in characteristic $p$

Abstract

In this article, we study the transcendence of special values of certain Goss type $L$-series at non-negative integers, which takes values in a function field of characteristic $p$. We show that, for a Drinfeld module $\varphi$ over $K$ and an Artin representation $\rho:G_K\to \operatorname{GL}_n(\overline{\mathbb{F}}_q)$, the twisted special value $L(\varphi^\vee,\rho,k)$ is transcendental over $K$ for every non-negative integer $k$. The proof uses the theory of Artin twists of Drinfeld modules, Taelman's regulators of $t$-modules, and the algebraic independence theorem of Gezmis and Namoijam for tractable coordinates of logarithms. As a consequence, we deduce the transcendence of the special $L$-value $L(\rho,k)$ for every positive integers $k$.