We investigate galaxy evolution in protoclusters using a semi-analytic model applied to the Millennium Simulation, scaled to a Planck cosmology. We show that the model reproduces the observed behaviour of the star formation history (SFH) both in protoclusters and the field. The rate of star formation peaks $\sim0.7\,{\rm Gyr}$ earlier in protoclusters than in the field and declines more rapidly afterwards. This results in protocluster galaxies forming significantly earlier: 80% of their stellar mass is already formed by $z=1.4$, but only 45% of the field stellar mass has formed by this time. The model predicts that field and protocluster galaxies have similar average specific star-formation rates (sSFR) at $z>3$, and we find evidence of an enhancement of star formation in the dense protoclusters at early times. At $z<3$, protoclusters have lower sSFRs, resulting in the disparity between the SFHs. We show that the stellar mass functions of protoclusters are top-heavy compared with the field due to the early formation of massive galaxies, and the disruption and merging of low-mass satellite galaxies in the main haloes. The fundamental cause of the different SFHs and mass functions is that dark matter haloes are biased tracers of the dark matter density field: the high density of haloes and the top-heavy halo mass function in protoclusters result in the early formation then rapid merging and quenching of galaxies. We compare our results with observations from the literature, and highlight which observables provide the most informative tests of galaxy formation.

14 pages, 9 figures, accepted for publication in MNRAS