On Stronger Versions of Brumer's Conjecture

Abstract

Let $k$ be a totally real number field and $L$ a CM-field such that $L/k$ is finite and abelian. In this paper, we study a stronger version of Brumer's conjecture that the Stickelberger element times the annihilator of the group of roots of unity in $L$ is in the Fitting ideal of the ideal class group of $L$, and also study the dual version. We mainly study the Teichmüller character component, and determine the Fitting ideal in a certain case. We will see that these stronger versions hold in a certain case. It is known that the stronger version (SB) does not hold in general. We will prove in this paper that the dual version (DSB) does not hold in general, either.