Non-liftability of Families of Abelian Varieties with Small $l$-adic Local System

TL;DR

This paper investigates the properties of families of abelian varieties with small l-adic local systems over curves in characteristic p, revealing non-liftability and establishing inequalities under certain liftability conditions.

Contribution

It demonstrates that such abelian schemes have a non-nef Hodge bundle and cannot be lifted to W_2(k), and proves an Arakelov-type inequality assuming liftability.

Findings

Abelian schemes with small l-adic local systems have non-nef Hodge bundles.

Such schemes cannot be lifted to W_2(k).

An Arakelov-type inequality is established under liftability assumptions.

Abstract

We study families of abelian varieties over smooth proper curves with small $l$-adic local system over characteristic $p$. We show that such abelian schemes have a non-nef Hodge bundle and cannot be lifted to $W_2(k)$. We also establish an Arakelov-type inequality for families of abelian varieties over smooth proper curves in characteristic $p$, assuming $W_2(k)$-liftability.