  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.
