An Arakelov inequality in characteristic p and upper bound of p-rank zero locus

In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus g ≥ 1 over characteristic p with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic curves of p-rank zero in a semi-stable family over characteristic p with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. The parallel results for smooth families of Abelian varieties over k with W₂-lifting assumption are also obtained.