1 INTRODUCTION  
2 SAMPLE PROPERTIES AND DATA ANALYSIS  
2.1 Sample selection  
2.2 Emission line measurements and black hole mass estimation  
2.3 X-ray Data: Chandra Deep Field-South  
2.4 X-ray Data: XMM-COSMOS  
2.5 X-ray spectral analysis  
2.6 Determining λ Edd  
2.7 Radio properties of the sample  
3 WHAT INFLUENCES THE PHYSICAL CONDITIONS OF THE CORONA?  
3.1 $L_{\mathrm{X}}$ and redshift  
3.2 $\boldsymbol{L}_{\mathrm{UV}}$, FWHM, M and λ BH Edd  
3.3 Radio loud sources  
4 DISCUSSION  
4.1 Accounting for degeneracies and biases  
4.2 Comparison with previous studies  
4.3 Wider implications of our results  
5 CONCLUSIONS