Abelian varieties over finite fields and their groups of rational points
Stefano Marseglia, Caleb Springer
TL;DR
The paper addresses how endomorphism rings and conductor data govern the groups of rational points on abelian varieties over finite fields, clarifying when these groups are determined by the endomorphism structure and when they can vary within an isogeny class. It develops a toolkit based on fractional ideals in étale algebras, Gorenstein/Cohen–Macaulay types, and the Deligne–Centeleghe–Stix correspondence to describe $A(\\\\mathbb{F}_{q^n})$ as quotients $End(A)/(1-\pi^n)End(A)$ and to analyze cyclicity and richness of isogeny classes via conductor ideals. The work provides a rigorous framework linking duality, self-duality, and polarity to endomorphism-ring data, with concrete criteria for when $A(\\\\mathbb{F}_{q^n})$ is uniquely determined by $\,End(A)$, when squarefree isogeny classes are cyclic or rich, and when abelian varieties fail to be self-dual. These results yield practical criteria, examples, and explicit constructions that sharpen the understanding of how algebraic structure governs the arithmetic of rational points in families of abelian varieties over finite fields.
Abstract
We study the groups of rational points of abelian varieties defined over a finite field $ \mathbb{F}_q$ whose endomorphism rings are commutative, or, equivalently, whose isogeny classes are determined by squarefree characteristic polynomials. When $\mathrm{End}(A)$ is locally Gorenstein, we show that the group structure of $A(\mathbb{F}_q)$ is determined by $\mathrm{End}(A)$. Moreover, we prove that the same conclusion is attained if $\mathrm{End}(A)$ has local Cohen-Macaulay type at most $ 2$, under the additional assumption that $A$ is ordinary or $q$ is prime. The result in the Gorenstein case is used to characterize squarefree cyclic isogeny classes in terms of conductor ideals. Going in the opposite direction, we characterize squarefree isogeny classes of abelian varieties with $N$ rational points in which every abelian group of order $N$ is realized as a group of rational points. Finally, we study when an abelian variety $A$ over $\mathbb{F}_q$ and its dual $A^\vee$ succeed or fail to satisfy several interrelated properties, namely $A\cong A^\vee$, $A(\mathbb{F}_q)\cong A^\vee(\mathbb{F}_q)$, and $\mathrm{End}(A)=\mathrm{End}(A^\vee)$. In the process, we exhibit a sufficient condition for $A\not\cong A^\vee$ involving the local Cohen-Macaulay type of $\mathrm{End}(A)$. In particular, such an abelian variety $A$ is not a Jacobian, or even principally polarizable.