Combination of phase modulation with coherent detection for low Earth orbit satellite-to-ground optical links is currently investigated to meet the current high-data-rate increasing demand. On the downlink, the signal carrier undergoes a large Doppler frequency shift which may in turn severely hinders demodulation and information recovery. We present a coherent receiver architecture which combines adaptive optics correction, to mitigate the atmospheric turbulence detrimental effect, with a digital carrier synchronization system. The latter is based on a phase-locked loop which ensures the compensation of the residual frequency shift that remains after preliminary coarse frequency offset correction. We describe the methodology followed to design such a digital phase-locked loop. As an example, we show that the digital PLL is able to compensate a residual frequency mismatch of 300MHz with a convergence time inferior to 1s which is reasonable compared to the duration of the satellite pass.