  We consider a population of globally coupled oscillators driven by common
noise. By applying the Ott-Antonsen ansatz and by averaging over the fast
oscillations, we obtain analytically tractable equations for the noisy
evolution of the order parameter. While noise always tends to synchronize the
oscillators, the coupling can act against synchrony if it is repulsive. For
identical oscillators, the fully synchronous state remains stable for small
enough repulsive coupling; moreover it is an absorbing state which always wins
over the asynchronous regime. For oscillators with a distribution of natural
frequencies, we report on a counter-intuitive effect of dispersion of the
oscillators frequencies at synchrony.
