Weil polynomials of abelian varieties over finite fields with many rational points

TL;DR

This paper characterizes the unique isogeny class of abelian varieties over finite fields with maximal rational points, describing its Weil polynomial and properties like ordinarity and cyclicity, especially in dimension 3.

Contribution

It proves the uniqueness of the class with maximal rational points for large finite fields and explicitly describes its Weil polynomial and properties.

Findings

Unique isogeny class with maximal rational points identified

Weil polynomial of the class explicitly described

Class is proven to be ordinary and cyclic outside specific primes

Abstract

We consider the finite set of isogeny classes of $g$-dimensional abelian varieties defined over the finite field $\mathbb{F}_q$ with endomorphism algebra being a field. We prove that the class within this set whose varieties have maximal number of rational points is unique, for any prime even power $q$ big enough and verifying mild conditions. We describe its Weil polynomial and we prove that the class is ordinary and cyclic outside the primes dividing an integer that only depends on $g$. In dimension $3$, we prove that the class is ordinary and cyclic and give explicitly its Weil polynomial, for any prime even power $q$.