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

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 $\varphi$ defined over $K$, we study the transcendence of special values of the Goss $L$-function attached to the abelian $t$-motive $M_{\varphi }$ of $\varphi$. Moreover, when $\varphi$ is a Drinfeld module of rank $r\ge 2$ 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 $M_{\varphi }$ at any positive integer is transcendental over $K$.

22 pages