An Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications

Abstract

We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a $G$-equivariant $L$-function $\Theta_{K/F}^E$ and prove an equivariant Tamagawa number formula for certain Euler-completed versions of its special value $\Theta_{K/F}^E(0)$. This generalizes Taelman's class number formula for the value $\zeta_F^E(0)$ of the Goss zeta function $\zeta_F^E$ associated to the pair $(F, E)$. Taelman's result is obtained from our result by setting $K=F$. As a consequence, we prove a perfect Drinfeld module analogue of the classical (number field) refined Brumer--Stark conjecture, relating a certain $G$-Fitting ideal of Taelman's class group $H(E/K)$ to the special value $\Theta_{K/F}^E(0)$ in question.