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

TL;DR

The paper investigates Weil-central isogeny classes of abelian varieties over finite fields, focusing on local cyclicity and the growth of rational-point groups under field extensions. Employing Honda--Tate theory, it establishes a concrete cyclicity criterion in terms of the Weil polynomial: an isogeny class is cyclic if and only if $f'(1)$ is coprime with $\\widehat{f(1)}$. It then analyzes how extensions preserve the Weil-central form, derives exact formulas for the growth of the $\ell$-part of $A(k)$, and provides a complete description of local cyclicity at a prime $\ell$, including explicit sets $S_{g,\ell}$ and conditions related to $\omega_{\ell}(q^{g})$. The results are illustrated with explicit examples, highlighting the practical implications for understanding cyclicity and growth in families of abelian varieties, with potential applications to cryptography and arithmetic statistics. Overall, the work clarifies how Weil-central geometry behaves under extensions and how local invariants control cyclicity and growth phenomena.

Abstract

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.

Theorems & Definitions (16)

• definition 1
• theorem 1: A. Giangreco, 2019, GIANGRECOMAIDANA2019139
• definition 2
• theorem 2
• remark 1
• lemma 1
• proof
• lemma 2
• proof
• lemma 3