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A possibility of the realization of the p-wave spin-singlet superconductivity($p$SS), whose gap function is odd both in momentum and in frequency, isinvestigated by solving the gap equation with the phenomenological interactionmediated by the antiferromagnetic spin fluctuation. The $p$SS is realizedprevailing over the d-wave singlet superconductivity ($d$SS) in the vicinity ofantiferromagnetic quantum critical pint (QCP) both on the paramagnetic and onthe antiferromagnetic sides. Off the QCP in the paramagnetic phase, however,the $d$SS with line-nodes is realized as \textit{conventional} anisotropicsuperconductivity. For the present $p$SS state, there is no gap in thequasiparticle spectrum everywhere on the Fermi surface due to its oddfrequency. These features can give a qualitative understanding of the anomalousbehaviors of NQR relaxation rate on CeCu$_2$Si$_2$ or CeRhIn$_5$ where theantiferromagnetism and superconductivity coexist on a microscopic level.Comment: 20 pages with 12 figures. To appear in J. Phys. Soc. Jpn. Vol. 72,  No. 1

Realization of Odd-Frequencyp-Wave Spin–Singlet Superconductivity Coexisting with Antiferromagnetic Order near Quantum Critical Point