An Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications

We fix data $(K\mathrm{/}F,E)$ consisting of a Galois extension $K\mathrm{/}F$ 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 ${\mathrm{\Theta }}_{K\mathrm{/}F}^E$ and prove an equivariant Tamagawa number formula for certain Euler-completed versions of its special value ${\mathrm{\Theta }}_{K\mathrm{/}F}^E(0)$. 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 $H(E\mathrm{/}K)$ to the special value ${\mathrm{\Theta }}_{K\mathrm{/}F}^E(0)$ in question.