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

TL;DR

This paper establishes an Arakelov inequality for semi-stable algebraic curve families in characteristic p, providing bounds on p-rank zero loci and extending results to Abelian varieties under lifting assumptions.

Contribution

It introduces a new Arakelov inequality in characteristic p and derives bounds on p-rank zero loci, extending to Abelian varieties with W_2-lifting.

Findings

Derived an upper bound for p-rank zero curves in families over characteristic p.

Extended inequality results to Abelian varieties with W_2-lifting.

Provided explicit relations involving genus and singular fibers.

Abstract

In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus $g\geq 1$ 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. An extension of the above results to smooth families of Abelian varieties over $k$ with $W_2$-lifting assumption is also included.