We present results from a new ultra-deep 400 ks Chandra observation of the SSA22 protocluster at z = 3.09. We have studied the X-ray properties of 234 z ~ 3 Lyman break galaxies (LBGs; protocluster and field) and 158 z = 3.09 Ly-alpha emitters (LAEs) in SSA22 to measure the influence of the high-density protocluster environment on the accretion activity of supermassive black holes (SMBHs) in these UV-selected star forming populations. We detect individually X-ray emission from active galactic nuclei (AGNs) in six LBGs and five LAEs; due to small overlap between the LBG and LAE source population, ten of these sources are unique. At least six and potentially eight of these sources are members of the protocluster. These sources have rest-frame 8-32 keV luminosities in the range of L_8-32 keV = (3-50) X 10^{43} ergs/s and an average observed-frame 2-8 keV to 0.5-2 keV band-ratio of ~0.8 (mean effective photon index of Gamma_eff = 1.1), suggesting significant absorption columns of N_H > 10^{22}-10^{24} cm^{-2}. We find that the fraction of LBGs and LAEs in the z = 3.09 protocluster harboring an AGN with L_8-32 keV > 3 X 10^{43} ergs/s is 9.5^{+12.7}_{-6.1}% and 5.1^{+6.8}_{-3.3}%, respectively. These AGN fractions are somewhat larger (by a mean factor of 6.1^{+10.3}_{-3.6}; significant at the 95% confidence level) than z ~ 3 sources found in lower-density field environments. Theoretical models imply that these results may be due to the presence of more actively growing and/or massive SMBHs in LBGs and LAEs within the protocluster compared to the field. Such a result is expected in a scenario where enhanced merger activity in the protocluster drives accelerated galaxy and SMBH growth at z > 2-3. (abridged)
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