In the framework of high-data-rate free-space optical low Earth orbit satellite-to-ground communication, we investigate, by means of a refined end-to-end numerical model of the link, the performance of a coherent receiver that combines an adaptive optics system and a specific digital receiver architecture. The design of a fine carrier recovery stage based on a phase-locked loop is presented and its performance is characterized. The end-to-end model includes the impact of atmospheric turbulence, adaptive optics correction, laser phase noise and of the frequency mismatch between the transmit and receiver lasers. The results show that adaptive optics coupled with classical digital phase-locked loop techniques can provide a reliable solution to the problem of carrier frequency and phase tracking in coherent satellite-to-ground optical links, after prior coarse frequency estimation. The phase-locked loop converges after a few hundreds of microseconds and accurately tracks the phase fluctuations. The residual amplitude fluctuations and laser phase noise are shown to be the dominant impairments for the link performance.