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 $K:=\mathbb{F}_q(θ)$. For a Drinfeld module $ϕ$ defined over $K$, we study the transcendence of special values of the Goss $L$-function attached to the abelian $t$-motive $M_ϕ$ of $ϕ$. Moreover, when $ϕ$ 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 $M_ϕ$ at any positive integer is transcendental over $K$.