Constancy of Newton polygons of $F$-isocrystals on Abelian varieties and isotriviality of families of curves

TL;DR

This paper proves that Newton polygons of convergent F-isocrystals remain constant on Abelian varieties over finite fields and uses this to show certain families of curves are isotrivial, with implications for their geometric structure.

Contribution

It establishes the constancy of Newton polygons for all convergent F-isocrystals on Abelian varieties and applies this to prove isotriviality of families of curves over these varieties.

Findings

Newton polygons are constant on Abelian varieties over finite fields.

Families of curves over Abelian varieties are isotrivial.

Studied isotriviality over simply connected projective varieties.

Abstract

We prove constancy of Newton polygons of all convergent $F$-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of projective smooth families of curves over Abelian varieties. We also study the isotriviality over simply connected projective smooth varieties.