Motivated by the recent observational and theoretical evidence that long gamma-ray bursts (GRBs) are likely associated with low metallicity, rapidly rotating massive stars, we examine the cosmological star formation rate (SFR) below a critical metallicity Zcrit ~ 1/10-1/5Zsolar, to estimate the event rate of high redshift long GRB progenitors. To this purpose, we exploit a galaxy formation scenario already successfully tested on a wealth of observational data on (proto)spheroids, Lyman break galaxies, Lyman α emitters, submm galaxies, quasars and local early-type galaxies. We find that the predicted rate of long GRBs amounts to about 300 events yr-1 sr-1, of which about 30 per cent occur at z >~ 6. Correspondingly, the GRB number counts well agree with the bright SWIFT data, without the need for an intrinsic luminosity evolution. Moreover, the above framework enables us to predict the properties of the GRB host galaxies. Most GRBs are associated with low-mass galaxy haloes MH <~ 1011Msolar, and effectively trace the formation of small galaxies in such haloes. The hosts are young, with age smaller than 5 × 107 yr, gas rich, but poorly extincted (AV <~ 0.1) because of their chemical immaturity; this also implies high specific SFR and quite extreme α-enhancement. Only the minority of hosts residing in large haloes with MH >~ 1012Msolar has larger extinction (AV ~ 0.7 - 1), SFRs exceeding 100Msolaryr-1 and can be detected at submm wavelengths. Most of the hosts have ultraviolet magnitudes in the range -20 <~ M1350 <~ -16, and Lyman α luminosity in the range 2 × 1040 <~ LLymanα <~ 2 × 1042 erg s-1. GRB hosts are thus tracing the faint end of the luminosity function of Lyman break galaxies and Lyman α emitters. Finally, our results imply that the population of `dark' GRBs occur mostly in faint hosts at high redshift, rather than in dusty hosts at low redshift.
