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Existence of simple non-cyclic abelian varieties over arbitrary finite fields and of a given dimension $g>1$
Alejandro J. Giangreco Maidana
https://arxiv.org/abs/2507.06916

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