We present an investigation into how well the properties of the accretion flow onto a supermassive black hole may be coupled to those of the overlying hot corona. To do so, we specifically measure the characteristic spectral index, Gamma, of a power-
law energy distribution, over an energy range of 2 to 10 keV, for X-ray selected, broad-lined radio-quiet AGN up to z~2 in COSMOS and E-CDF-S. We test the previously reported dependence between Gamma and black hole mass, FWHM and Eddington ratio using a sample of AGN covering a broad range in these parameters based on both the Mg ii and H-alpha emission lines with the later afforded by recent near infrared spectroscopic observations using Subaru/FMOS. We calculate the Eddington ratios, lambda_Edd, for sources where a bolometric luminosity (L_Bol) has been presented in the literature, based on SED fitting, or, for sources where these data do not exist, we calculate L_Bol using a bolometric correction to the LX, derived from a relationship between the bolometric correction, and LX/L3000. From a sample of 69 X-ray bright sources (>250 counts), where Gamma can be measured with greatest precision, with an estimate of L_Bol, we find a statistically significant correlation between Gamma and lambda_Edd, which is highly significant with a chance probability of 6.59x10^-8. A statistically significant correlation between Gamma and the FWHM of the optical lines is confirmed, but at lower significance than with lambda_Edd indicating that lambda_Edd is the key parameter driving conditions in the corona. Linear regression analysis reveals that Gamma=(0.32+/-0.05)log10 lambda_Edd+(2.27+/-0.06) and Gamma=(-0.69+/-0.11)log10(FWHM/km/s)+(4.44+/-0.42). Our results on Gamma-lambda_Edd are in very good agreement with previous results. (ABRIDGED)
