Stickelberger splitting in the K-theory of number fields

I defined the Stickelberger splitting map in the case of abelian extensions $F/\mathbb{Q}$ in my Ph.D thesis in 1990. The construction used the classical Stickelberger's theorem. For abelian extensions $F/K,$ with an arbitrary totally real base field $K,$ the construction cannot be generalized since Brumer's conjecture (the analogue of  Stickelberger's theorem) is not proved yet at that level of generality. In my talk I will describe results of a joint paper with Cristian Popescu of  the construction of the Stickelberger splitting map for abelian CM extensions  $F/K$ under the assumption that the first Stickelberger elements annihilate  the Quillen $K$-groups $K_2 ({\mathcal O}_{F_{l^k}})$ for the Iwasawa tower $F_{l^k} := F(\mu_{l^k})$, for $k \geq 1.$ Recently Cristian Popescu has given examples  of CM abelian extensions $F/K$ of general totally real base-fields $K$ for which the first Stickelberger elements annihilate $K_2 ({\mathcal O}_{F_{l^k}})_l$ for all $k \geq 1$. As a consequence, the Stickelberger splitting map leads to annihilation results for the groups of divisible elements in even $K$-groups of $F$ as predicted by the generalized Coates-Sinnott conjecture.