Some elementary theorems about divisibility of 0-cycles on abelian varieties defined over finite fields

TL;DR

This paper investigates the divisibility properties of 0-cycles on abelian varieties over finite fields, revealing new elementary theorems that suggest potential divisibility patterns not observed over other fields.

Contribution

The paper establishes new elementary theorems about the divisibility of 0-cycles on abelian varieties over finite fields, highlighting differences from other fields.

Findings

Degree of $L^g$ divisible by $g!$ over any field

Divisibility properties differ over finite fields

Potential for new divisibility results over finite fields

Abstract

If $X$ is an abelian variety over a field and $L$ is an invertible sheaf, we know that the degree of the 0-cycle $L^g$ is divisible by $g!$. As a 0-cycle, it is not, even over a field of cohomological dimension 1. But we show that over a finite field there is perhaps some hope.