Maximal abelian varieties over finite fields and cyclicity

Elena Berardini, Alejandro J. Giangreco Maidana

Published 2021-01-29Version 1

Let $\mathcal{M}_g^0(q)$ denote the isogeny class (if it is unique) of $g$-dimensional abelian varieties defined over $\mathbb{F}_q$ with maximal number of rational points among those with endomorphism algebra being a field. We describe a polynomial $h_g(t,X)$ such that for every even power $q$ of a prime and verifying mild conditions, $h_g(t,\sqrt{q})$ is the Weil polynomial of $\mathcal{M}_g^0(q)$. It turns out that in this case, $\mathcal{M}_g^0(q)$ is ordinary and cyclic outside the primes dividing an integer $N_g$ that only depends on $g$. We also give explicitly $h_3(t,X)$ and prove that $\mathcal{M}_3^0(q)$ is ordinary and cyclic.