Fitting ideals of class groups for CM abelian extensions

TL;DR

This paper determines the Fitting ideal of the minus component of the $T$-ray class group in CM abelian extensions, assuming the equivariant Tamagawa number conjecture, and relates it to Stickelberger elements.

Contribution

It provides an explicit description of the Fitting ideal for class groups in CM extensions under certain conjectural assumptions, extending previous results.

Findings

Fitting ideal of the minus component is explicitly determined.

A necessary and sufficient condition for Stickelberger elements to lie in the Fitting ideal.

Results depend on the validity of the equivariant Tamagawa number conjecture.

Abstract

Let $K/k$ be a finite abelian CM-extension and $T$ a suitable finite set of finite primes of $k$. In this paper, 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.