In the framework of optical high data rate satellite-to-ground links, we investigate the performance of a coherent receiver which combines an adaptive optics system and a digital carrier synchronization technique. We propose a digital carrier synchronization system based on a phase-locked loop and use end-to-end simulations to characterize the impact of the atmospheric turbulence in the case of a representative low Earth orbit satellite-to-ground transmission link. The results show that classical digital phase-locked loop techniques can be a reliable solution to accurately compensate for the Doppler frequency shift between the incoming signal and the local oscillator at the ground station, in the presence of a preliminary coarse frequency estimation. Adaptive optics compensation enables the loop to lock after a few milliseconds and to accurately track the phase fluctuations. The bit-error-rate performance is not degraded by the digital loop but suffers from the residual signal amplitude fluctuations.