arXiv subjects

Alejandro J. Giangreco-Maidana

Publications and source records attributed to Alejandro J. Giangreco-Maidana.

3 recordsLinked to original sources

arXiv2025

Some arithmetic properties of Weil polynomials of the form $t^{2g}+at^g+q^g$

An isogeny class $\mathcal{A}$ of abelian varieties defined over finite fields is said to be "cyclic" if every variety in $\mathcal{A}$ has a cyclic group of rational points. In this paper we study the local cyclicity of Weil-central isogeny classes of abelian varieties, i.e. those with Weil polynomials of the form $f_\mathcal{A}(t)=t^{2g}+at^g+q^g$, as well as the local growth of the groups of rational points of the varieties in $\mathcal{A}$ after finite field extensions. We exploit the criterion: an isogeny class $\mathcal{A}$ with Weil polynomial $f$ is cyclic if and only if $f'(1)$ is coprime with $f(1)$ divided by its radical.

math.NT↗

arXiv2018

Local cyclicity of isogeny classes of abelian varieties defined over finite fields

For a prime number $\ell$, an isogeny class $\mathcal{A}$ of abelian varieties is called $\ell$-cyclic if every variety in $\mathcal{A}$ have a cyclic $\ell$-part of its group of rational points. More generally, for a finite set of prime numbers $\mathcal{S}$, $\mathcal{A}$ is said to be $\mathcal{S}$-cyclic if it is $\ell$-cyclic for every $\ell\in\mathcal{S}$. We give lower and upper bounds on the fraction of $\mathcal{S}$-cyclic $g$-dimensional isogeny classes of abelian varieties defined over the finite field $\mathbb{F}_{q}$, when $q$ tends to infinity.

math.AG↗

arXiv2018

On the cyclicity of the rational points group of abelian varieties over finite fields

We propose a simple criterion to know if an abelian variety $A$ defined over a finite field $\mathbb{F}_q$ is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring End$_{\mathbb{F}_q}(A)$. We also provide a criterion to know if an isogeny class is cyclic, i.e., all its varieties are cyclic; this criterion is based on the characteristic polynomial of the isogeny class. We find some asymptotic lower bounds on the fraction of cyclic $\mathbb{F}_q$-isogeny classes among certain families of them, when $q$ tends to infinity. Some of these bounds require an additional hypothesis. In the case of surfaces, we prove that this hypothesis is achieved and, over all $\mathbb{F}_q$-isogeny classes with endomorphism algebra being a field and where $q$ is an even power of a prime, we prove that the one with maximal number of rational points is cyclic and ordinary.

math.AG↗