Algebra Number Theory 16 (9), 2215-2264, (2022) DOI: 10.2140/ant.2022.16.2215
Joseph Ferrara, Nathan Green, Zach Higgins, Cristian D. Popescu
KEYWORDS: Drinfeld modules, motivic L-functions, equivariant Tamagawa number formula, Brumer–Stark conjecture, 11F80, 11G09, 11M38
We fix motivic data consisting of a Galois extension of characteristic global fields with arbitrary abelian Galois group and a Drinfeld module defined over a certain Dedekind subring of . For this data, we define a -equivariant motivic -function and prove an equivariant Tamagawa number formula for appropriate Euler product completions of its special value . This generalizes to an equivariant setting the celebrated class number formula proved by Taelman in 2012 for the value of the Goss zeta function associated to the pair . (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 . As a notable consequence, we prove a perfect Drinfeld module analogue of the classical (number field) refined Brumer–Stark conjecture, relating a certain -Fitting ideal of Taelman’s class group to the special value in question. In upcoming work, these results will be extended to the category of -modules and used in developing an Iwasawa theory for Taelman’s class groups in Carlitz cyclotomic towers.