  Co nuclear-quadrupole-resonance (NQR) measurements were performed on various
bilayered hydrate cobaltate Na_x(H_3O)_zCoO_2\cdot yH_2O with different values
of the superconducting and magnetic-ordering temperatures, T_c and T_M,
respectively. From measurements of the temperature and sample dependence of the
NQR frequency, it was revealed that the NQR frequency is changed by the change
of the electric field gradient (EFG) along the c axis \nu_{zz} rather than the
asymmetry of EFG within the ab-plane. In addition, it is considered that the
change of \nu_{zz} is gaverned mainly by the trigonal distortion of the CoO_2
block layers along the c axis, from the relationships between \nu_{zz} and the
various physical parameters. We found the tendency that samples with \nu_{zz}
larger than 4.2 MHz show magnetic ordering, whereas samples with lower \nu_{zz}
show superconductivity. We measured the nuclear spin-lattice relaxation rate
1/T_1 in these samples, and found that magnetic fluctuations depend on samples.
The higher-\nu_{zz} sample has stronger magnetic fluctuations at T_c. From the
relationship between \nu_{zz} and T_c or T_M, we suggest that the NQR frequency
can be regarded as a tuning parameter to determine the ground state of the
system, and develop the phase diagram using \nu_{zz}. This phase diagram shows
that the highest-T_c sample is located at the point where T_M is considered to
be zero, which suggests that the superconductivity is induced by quantum
critical fluctuations. We strongly advocate that the hydrate cobaltate
superconductor presents an example of the magnetic-fluctuation-mediated
superconductivity argued in the heavy-fermion compounds. The coexistence of
superconductivity and magnetism observed in the sample with the highest
\nu_{zz} is also discussed on the basis of the results of our experiments.

Superconductivity;Strongly Correlated Electrons