Anderson modules and their equivarient L-functions

Let $K$ be a finite extension of the rational field $\mathbb{F}_q(\theta)$ and $\mathcal{O}_K$ be the ring of integers of $K$. Let $E$ be an Anderson $t$-module defined over $\mathcal{O}_K$ and $L/K$ be an abelian extension of Galois group $G$. We obtain a $G$-equivariant class formula for the canonical $z$-deformation of $E/\mathcal{O}_L$ which relates an equivariant regulator of Taelman’s units to some special value of equivariant Goss $L$-function. Under mild conditions over the class module of Taelman, it allows us to get a $G$-equivariant class formula for $E/\mathcal{O}_L$.