On the mod $p$ Lannes-Zarati homomorphism

The mod $2$ Lannes-Zarati homomorphism was constructed in \cite{Lan-Zar87}, which is considered as a graded associated version of the mod $2$ Hurewicz map in the $E_2$-term of Adams spectral sequence. The map is studied by many authors such as Lannes-Zarati \cite{Lan-Zar87}, H\horn{u}ng \cite{Hung97}, \cite{Hung2001}, \cite{Hung2003}, H\horn{u}ng et. al. \cite{Hung.et.al2014}, Ch\horn{o}n-Tri\'\^et \cite{Chon.Triet2014}. In this paper, we construct an analogue $\varphi_s$ for $p$ odd, and we also investigate the behavior of this map for $s\leq 3$.