An equivariant Tamagawa number formula for Drinfeld modules and applications

Ferrara, Joseph; Green, Nathan; Higgins, Zach; Popescu, Cristian D.

doi:10.2140/ant.2022.16.2215

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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

DOI: 10.2140/ant.2022.16.2215

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 a Drinfeld module $E$ defined over a certain Dedekind subring of $F$ . For this data, we define a $G$ -equivariant motivic $L$ -function $Θ_{K ∕ F}^{E}$ and prove an equivariant Tamagawa number formula for appropriate Euler product completions of its special value $Θ_{K ∕ F}^{E} (0)$ . This generalizes to an equivariant setting the celebrated class number formula proved by Taelman in 2012 for the value $ζ_{F}^{E} (0)$ of the Goss zeta function $ζ_{F}^{E}$ associated to the pair $(F, E)$ . (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 $H (E ∕ K)$ to the special value $Θ_{K ∕ F}^{E} (0)$ 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.

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Keywords

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

Mathematical Subject Classification

Primary: 11F80, 11G09, 11M38

Milestones

Received: 1 August 2021

Revised: 5 October 2021

Accepted: 12 November 2021

Published: 19 December 2022

Authors

Joseph Ferrara
Department of Mathematics University of California, San Diego La Jolla, CA United States

Nathan Green
Department of Mathematics University of California, San Diego La Jolla, CA United States

Zach Higgins
Department of Mathematics University of California, San Diego La Jolla, CA United States

Cristian D. Popescu
Department of Mathematics University of California, San Diego La Jolla, CA United States

Vol. 16, No. 9, 2022