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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
Abstract

We fix motivic data (KF,E) consisting of a Galois extension KF 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 ΘKFE and prove an equivariant Tamagawa number formula for appropriate Euler product completions of its special value ΘKFE(0). This generalizes to an equivariant setting the celebrated class number formula proved by Taelman in 2012 for the value ζFE(0) of the Goss zeta function ζFE 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(EK) to the special value ΘKFE(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.

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