Fitting ideals of class groups for CM abelian extensions

Atsuta, Mahiro; Kataoka, Takenori

doi:10.2140/ant.2023.17.1901

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Fitting ideals of class groups for CM abelian extensions

Mahiro Atsuta and Takenori Kataoka

Vol. 17 (2023), No. 11, 1901–1924

DOI: 10.2140/ant.2023.17.1901

Abstract

Let $K$ be a finite abelian CM-extension of a totally real field $k$ and $T$ a suitable finite set of finite primes of $k$ . We determine the Fitting ideal of the minus component of the $T$ -ray class group of $K$ , except for the $2$ -component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in that Fitting ideal.

Keywords

class groups, Fitting ideals, CM-fields, equivariant Tamagawa number conjecture

Mathematical Subject Classification

Primary: 11R29

Milestones

Received: 13 December 2021

Revised: 25 December 2022

Accepted: 13 February 2023

Published: 3 October 2023

Authors

Mahiro Atsuta
Institute for Mathematics and Computer Science Tsuda University Tokyo Japan

Takenori Kataoka
Department of Mathematics, Faculty of Science Division II Tokyo University of Science Tokyo Japan

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Vol. 17, No. 11, 2023