An equivariant Tamagawa number formula for Drinfeld modules and applications

Abstract

We fix motivic 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 motivic $L$-function ${\mathrm{\Theta }}_{K∕F}^E$ and prove an equivariant Tamagawa number formula for appropriate Euler product completions of its special value ${\mathrm{\Theta }}_{K∕F}^E(0)$. This generalizes to an equivariant setting the celebrated class number formula proved by Taelman in 2012 for the value ${\zeta }_F^E(0)$ of the Goss zeta function ${\zeta }_F^E$ associated to the pair $(F,E)$. (See also Mornev’s 2018 work for a generalization in a very different, nonequivariant direction.) We refine and adapt Taelman’s techniques to the general equivariant setting and recover his precise formula in the particular case $K=F$. As a notable 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 ${\mathrm{\Theta }}_{K∕F}^E(0)$ in question. In upcoming work, these results will be extended to the category of $t$-modules and used in developing an Iwasawa theory for Taelman’s class groups in Carlitz cyclotomic towers.