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

Abstract: Let $\mathbb{F}<em>q$ 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 $M</em>{\phi}$ 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 $K$.