Not yet reviewed by Pith; the record is open.

The pressure-temperature-phase diagram of the new heavy-fermion superconductor CeAu$_2$Si$_2$ is markedly different from those studied previously. Indeed, superconductivity emerges, not on the verge, but deep inside the magnetic phase. In this context, we have carried out ac-calorimetry, resistivity and thermoelectric power measurements on a CeAu$_2$Si$_2$-single crystal under high-pressure. The principal novelties of this experiment are the observation of a new transition line, presumably a magnetic structure rearrangement at $T_{\mathrm{M}}^{\mathrm{mod}}$ inside the antiferromagnetic phase, and the occurrence of quantum critical behavior in resistivity linked to superconductivity. Strong non-Fermi-liquid behavior is observed around the maximum of superconductivity and enhanced scattering rates are observed close to both the emergence and the maximum of superconductivity. Intriguingly, $T_{\mathrm{M}}^{\mathrm{mod}}$ almost coincides with the onset of the superconducting transition over a broad pressure range, where mysteriously Tc increases with the strengthening of magnetism. A comparison of the features in CeAu$_2$Si$_2$ and its parent compounds CeCu$_2$Si$_2$ and CeCu$_2$Ge$_2$ plotted as function of the unit-cell volume leads us to promote that critical fluctuations of a valence crossover play a crucial role in the superconducting paring mechanism. Our study reveals new intriguing features of magnetism and illustrates a complex interplay between magnetism, quantum criticality and superconductivity.