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In this paper we show an Arakelov inequality for semi-stable families ofalgebraic curves of genus $g\geq 1$ over characteristic $p$ with nontrivialKodaira-Spencer maps. We apply this inequality to obtain an upper bound of thenumber of algebraic curves of $p-$rank zero in a semi-stable family overcharacteristic $p$ with nontrivial Kodaira-Spencer map in terms of the genus ofa general closed fiber, the genus of the base curve and the number of singularfibres. An extension of the above results to smooth families of Abelianvarieties over $k$ with $W_2$-lifting assumption is also included.

An Arakelov Inequality in Characteristic p and Upper Bound of p-Rank Zero Locus