Stickelberger annihilators of logarithmic class groups

For any odd prime number $ell$ and any abelian number field F containing the $ell$-th roots of unity, we show that the Stickelberger ideal annihilates the imaginary component of the $ell$-group of logarithmic classes and that its reflection annihilates the real componen of the Bertrandias-Payan module. As a consequence we obtain a very simple proof of annihilation results for the so-called wild {e}tale $ell$-kernels of F .