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.
Keywords
Drinfeld modules, motivic $L$-functions, equivariant
Tamagawa number formula, Brumer–Stark conjecture