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The MUSE Hubble Ultra Deep Field Survey. XV. The mean rest-UV spectra of Ly-alpha emitters at z>3

We investigate the ultraviolet (UV) spectral properties of faint Lyman-$α$ emitters (LAEs) in the redshift range 2.9<z<4.6 and provide material to prepare future observations of the faint Universe. We use data from the MUSE Hubble Ultra Deep Survey to construct mean rest-frame spectra of continuum-faint (median M$_{UV}$ of -18 and down to M$_{UV}$ of -16), low stellar mass (median value of $10^{8.4}$ and down to $10^{7}M_{\odot}$) LAEs at redshift z>3. We compute various averaged spectra of LAEs sub-sampled on the basis of their observational (e.g., Ly$α$ strength, UV magnitude and spectral slope) and physical (e.g., stellar mass and star-formation rate) properties. We search for UV spectral features other than Ly$α$, such as higher-ionization nebular emission lines and absorption features. We successfully observe the OIII]1666 and CIII]909 collisionally excited emission lines and the HeII1640 recombination feature, as well as the resonant CIV1550 doublet either in emission or P-Cygni. We compare the observed spectral properties of the different mean spectra and find the emission lines to vary with the observational and physical properties of the LAEs. In particular, the mean spectra of LAEs with larger Ly$α$ equivalent widths, fainter UV magnitudes, bluer UV spectral slopes and lower stellar masses show the strongest nebular emission. The line ratios of these lines are similar to those measured in the spectra of local metal-poor galaxies, while their equivalent widths are weaker compared to the handful of extreme values detected in individual spectra of z>2 galaxies. This suggests that weak UV features are likely ubiquitous in high z, low-mass and faint LAEs. We publicly release the stacked spectra as they can serve as empirical templates for the design of future observations, such as those with the James Webb Space Telescope and the Extremely Large Telescope.