On the Brumer-Stark Conjecture

Understand

Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field.

• Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions.
• The Brumer-Stark conjecture states that the Stickelberger element $\Theta^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $\text{Cl}^T(H)$.
• We prove this conjecture away from $p=2$, that is, after tensoring with $\mathbf{Z}[1/2]$.