  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 .
