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Here, we study the X-ray and radio properties of \u223c4000 LAEs at 2.2 &amp;lt; $z$ &amp;lt; 6 from the SC4K survey in the COSMOS field. We detect 254 ($6.8{\\rm{ per\\ cent}} \\pm 0.4{\\rm{ per\\ cent}}$) LAEs individually in the X-rays (S\/N\u2009&amp;gt;\u20093) with an average luminosity of $\\rm 10^{44.31\\pm 0.01}\\, erg\\, s^{-1}$ and average black hole accretion rate (BHAR) of $\\rm 0.72 \\pm 0.01$\u2009M\u2299\u2009yr\u22121, consistent with moderate to high accreting active galactic neuclei (AGNs). We detect 120 sources in deep radio data (radio AGN fraction of $3.2{\\rm{ per\\ cent}} \\pm 0.3{\\rm{ per\\ cent}}$). The global AGN fraction ($\\rm 8.6{\\rm{ per\\ cent}} \\pm 0.4{\\rm{ per\\ cent}}$) rises with Ly\u2009\u03b1 luminosity and declines with increasing redshift. For X-ray-detected LAEs, Ly\u2009\u03b1 luminosities correlate with the BHARs, suggesting that Ly\u2009\u03b1 luminosity becomes a BHAR indicator. Most LAEs ($93.1{\\rm{ per\\ cent}} \\pm 0.6{\\rm{ per\\ cent}}$) at 2 &amp;lt; $z$ &amp;lt; 6 have no detectable X-ray emission (BHARs\u2009&amp;lt;\u20090.017\u2009M\u2299 yr\u22121). The median star formation rate (SFR) of star-forming LAEs from Ly\u2009\u03b1 and radio luminosities is $7.6^{+6.6}_{-2.8}$\u2009M\u2299\u2009yr\u22121. The black hole to galaxy growth ratio (BHAR\/SFR) for LAEs is &amp;lt;0.0022, consistent with typical star-forming galaxies and the local BHAR\/SFR relation. We conclude that LAEs at 2 &amp;lt; $z$ &amp;lt; 6 include two different populations: an AGN population, where Ly\u2009\u03b1 luminosity traces BHAR, and another with low SFRs which remain undetected in even the deepest X-ray stacks but is detected in the radio stacks.<\/jats:p>","DOI":"10.1093\/mnras\/staa476","type":"journal-article","created":{"date-parts":[[2020,2,17]],"date-time":"2020-02-17T15:13:35Z","timestamp":1581952415000},"page":"3341-3362","source":"Crossref","is-referenced-by-count":17,"title":["The X-ray and radio activity of typical and luminous Ly\u2009\u03b1 emitters from z \u223c 2 to z \u223c 6: evidence for a diverse, evolving population"],"prefix":"10.1093","volume":"493","author":[{"given":"Jo\u00e3o","family":"Calhau","sequence":"first","affiliation":[{"name":"Department of Physics, Lancaster University, Lancaster LA1 4YB, UK"}]},{"given":"David","family":"Sobral","sequence":"first","affiliation":[{"name":"Department of Physics, Lancaster University, Lancaster LA1 4YB, UK"}]},{"given":"S\u00e9rgio","family":"Santos","sequence":"first","affiliation":[{"name":"Department of Physics, Lancaster University, Lancaster LA1 4YB, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2871-127X","authenticated-orcid":false,"given":"Jorryt","family":"Matthee","sequence":"first","affiliation":[{"name":"Department of Physics, ETH Z\u00fcrich, Wolfgang-Pauli Strasse 27, CH-8093 Z\u00fcrich, Switzerland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0943-0694","authenticated-orcid":false,"given":"Ana","family":"Paulino-Afonso","sequence":"first","affiliation":[{"name":"Department of Physics, Lancaster University, Lancaster LA1 4YB, UK"},{"name":"CENTRA \u2013 Centro de Astrof\u00edsica e Gravita\u00e7\u00e3o, Instituto Superior T\u00e9cnico, Av. 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