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We present spectroscopic observations obtained with the ESO Very LargeTelecope (VLT) of seven candidate Ly-alpha emitting galaxies in the field ofthe radio quiet Q1205-30 at z=3.04 previously detected with deep narrow bandimaging. Based on equivalent widths and limits on line ratios we confirm thatall seven objects are Ly-alpha emitting galaxies. Deep images also obtainedwith the VLT in the B and I bands show that five of the seven galaxies havevery faint continuum fluxes (I(AB) approx. 26.8 and B(AB) approx. 27.3). Thestar formation rates of these seven galaxies estimated from the rest-frame UVcontinuum around 2000AA, as probed by the I-band detections, as well as fromthe Ly-alpha luminosities, are 1-4 M_sun yr^{-1}. This is 1-3 orders ofmagnitude lower than for other known populations of star-forming galaxies atsimilar redshifts (the Lyman-Break galaxies and the sub-mm selected sources).The inferred density of the objects is high, 16+-4 per arcmin^2 per unitredshift. This is consistent with the integrated luminosity function forLyman-Break galaxies down to R=27 if the fraction of Ly-alpha emitting galaxiesis approx. 70% at the faint end of the luminosity function. However, if thisfraction is 20% as reported for the bright end of the luminosity function thenthe space density in this field is significantly larger (by a factor of 3.5)than expected from the luminosity function for Lyman-Break galaxies in theHDF-North. This would be an indication that at least some radio quiet QSOs athigh redshift reside in overdense environments or that the faint end slope ofthe high redshift luminosity function has been underestimated. Theseobservations show that Ly-alpha emission is an efficient method by which toprobe the faint end of the luminosity function at high redshifts.Comment: 10 pages, 5 figures; Accepted for publication in the A&A main journa

Probing the faint end of the Galaxy luminosity function at $\vec{z}=$ 3 with Lyαemission