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An equivariant Tamagawa number formula for Drinfeld modules and applications

### Joseph Ferrara, Nathan Green, Zach Higgins and Cristian D. Popescu

Vol. 16 (2022), No. 9, 2215–2264
##### Abstract

We fix motivic data $\left(K∕F,E\right)$ 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 ${\Theta }_{K∕F}^{E}$ and prove an equivariant Tamagawa number formula for appropriate Euler product completions of its special value ${\Theta }_{K∕F}^{E}\left(0\right)$. This generalizes to an equivariant setting the celebrated class number formula proved by Taelman in 2012 for the value ${\zeta }_{F}^{E}\left(0\right)$ of the Goss zeta function ${\zeta }_{F}^{E}$ associated to the pair $\left(F,E\right)$. (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 $H\left(E∕K\right)$ to the special value ${\Theta }_{K∕F}^{E}\left(0\right)$ 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.

##### Keywords
Drinfeld modules, motivic $L$-functions, equivariant Tamagawa number formula, Brumer–Stark conjecture
##### Mathematical Subject Classification
Primary: 11F80, 11G09, 11M38