Existence of simple non-cyclic abelian varieties over arbitrary finite fields and of a given dimension $g>1$

TL;DR

The paper proves that over any finite field, there exist simple abelian varieties of dimension greater than one whose groups of rational points are not cyclic, extending previous results known for elliptic curves.

Contribution

It establishes the existence of simple non-cyclic abelian varieties of any dimension greater than one over arbitrary finite fields, a significant generalization of earlier elliptic curve results.

Findings

No finite field has all simple abelian varieties of dimension >1 with cyclic rational points.

Existence of simple abelian varieties with non-cyclic groups over any finite field.

Extends known elliptic curve results to higher-dimensional abelian varieties.

Abstract

Vl{\u a}du{\c t} characterized in 1999 the set of finite fields $k$ such that all elliptic curves defined over $k$ have a cyclic group of rational points. Under the conjecture of infinitely many Mersenne primes, this set is infinite. In these notes we prove that there is no a finite field $k$ such that all the simple abelian varieties defined over $k$ of dimension $g>1$ have a cyclic group of rational points.