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2022 An equivariant Tamagawa number formula for Drinfeld modules and applications
Joseph Ferrara, Nathan Green, Zach Higgins, Cristian D. Popescu
Algebra Number Theory 16(9): 2215-2264 (2022). DOI: 10.2140/ant.2022.16.2215


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.


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Joseph Ferrara. Nathan Green. Zach Higgins. Cristian D. Popescu. "An equivariant Tamagawa number formula for Drinfeld modules and applications." Algebra Number Theory 16 (9) 2215 - 2264, 2022.


Received: 1 August 2021; Revised: 5 October 2021; Accepted: 12 November 2021; Published: 2022
First available in Project Euclid: 18 January 2023

Digital Object Identifier: 10.2140/ant.2022.16.2215

Primary: 11F80 , 11G09 , 11M38

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

Rights: Copyright © 2022 Mathematical Sciences Publishers


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Vol.16 • No. 9 • 2022
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