An Equivariant Tamagawa Number Formula for t-Modules and Applications

TL;DR

This paper establishes an equivariant Tamagawa number formula for $t$-modules over global function fields, extending previous work and connecting special $L$-values to class modules, with applications to conjectures analogous to classical number theory.

Contribution

The authors prove an equivariant Tamagawa number formula for $t$-modules, extending prior results from Drinfeld modules and establishing analogues of classical conjectures in the function field setting.

Findings

Proved an equivariant Tamagawa number formula for $t$-modules.

Established a $t$-module analogue of the Refined Brumer-Stark Conjecture.

Derived formulas for special values of $L$-functions at positive integers.

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 ableian $t$-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 previous work of the authors 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 previous results of the authors 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 $t$-motive analogue $H(E/K)$ of Taelman's class modules 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 $m\in\Bbb Z_{\geq 0}$, 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 $H(E(m)/\mathcal O_K)$ 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.