This work is part of a theoretical project which aims to unravel the mechanism at work in the high temperature cuprate superconductors. In the recent article [1] we have studied the role of spin fluctuations, and namely of the collective spin mode, for the superconducting pairing and numerous electronic anomalies observed in cuprates. This mode strongly coupled to the electrons (which develops in the vicinity of the antiferromagnetic wave vector and has an anomalous downward dispersion) was first predicted theoretically [2] and then observed by neutrons [3]. More...