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We measure the evolution of the correlation between black hole mass and hostspheroid velocity dispersion over the last 6 billion years, by studying threecarefully selected samples of active galaxies at z=0.57, z=0.36 and z<0.1. Forall three samples, virial black hole masses are consistently estimated usingthe line dispersion of H$\beta$ and the continuum luminosity at 5100A or Halphaline luminosity, based on our cross calibration of the broad line regionsize-luminosity relation. For the z=0.57 sample, new stellar velocitydispersions are measured from high signal-to-noise ratio spectra obtained atthe Keck Telescope, while for the two lower redshift samples they are compiledfrom previous works. Extending our previous result at z=0.36, we find an offsetfrom the local relation, suggesting that for fixed M_{BH}, distant spheroidshave on average smaller velocity dispersions than local ones. The measuredoffset at z=0.57 is d log sigma_{*}=0.12 \pm 0.05\pm 0.06 (or d log M_{BH}=0.50\pm 0.22\pm 0.25), i.e. d log M_{BH} = (3.1\pm1.5)\log (1+z) + 0.05\pm0.21.This is inconsistent with a tight and non-evolving universal M-sigma relationat the 95%CL

Cosmic Evolution of Black Holes and Spheroids. III. The M-sigma relation in the last six billion years