An equivariant Tamagawa number formula for abelian $t$-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 an abelian $t$-module $E$, defined over a certain Dedekind subring of $F$. For this data, one can define a $G$-equivariant motivic $L$-function ${\Theta }_{K/F}^E$. We refine the techniques developed in [Algebra Number Theory 16 (2022), 2215–2264] and prove an equivariant Tamagawa number formula for appropriate Euler product completions of the special value ${\Theta }_{K/F}^E(0)$ of this equivariant $L$-function. This extends our results in op. cit. from the Drinfeld module setting to the $t$-module setting. As a first notable consequence, we prove a $t$-module analogue of the classical (number field) refined Brumer–Stark conjecture, relating a certain $G$-Fitting ideal of the $t$-module analogue of Taelman's class modules [Ann. of Math. (2) 175 (2012), 369–391] to the special value ${\Theta }_{K/F}^E(0)$ in question. As a second consequence, we prove formulas for the values ${\Theta }_{K/F}^E(m)$, at all positive integers $m\in {\mathbb{Z}}_{\ge 0}$, when $E$ is a Drinfeld module. This, in turn, implies a Drinfeld module analogue of the classical refined Coates–Sinnott conjecture relating ${\Theta }_{K/F}^E(m)$ to the Fitting ideals of certain Carlitz twists of Taelman's class modules, suggesting a strong analogy between these twists and the even Quillen $K$-groups of a number field. In an upcoming paper, these consequences will be used to develop an Iwasawa theory for the $t$-module analogues of Taelman's class modules.