Computing square-free polarized abelian varieties over finite fields

We give algorithms to compute isomorphism classes of ordinary abelian varieties defined over a finite field $\mathbb{F}_q$ whose characteristic polynomial (of Frobenius) is square-free and of abelian varieties defined over the prime field $\mathbb{F}_p$ whose characteristic polynomial is square-free and does not have real roots. In the ordinary case we are also able to compute the polarizations and the group of automorphisms (of the polarized variety) and, when the polarization is principal, the period matrix.Comment: accepted by Math. Comp. major revision: added computation of the group of points; examples have been exported on the rep

Super-multiplicativity of ideal norms in number fields

In this article we study inequalities of ideal norms. We prove that in a subring $R$ of a number field every ideal can be generated by at most $3$ elements if and only if the ideal norm satisfies $N(IJ) \geq N(I)N(J)$ for every pair of non-zero ideals $I$ and $J$ of every ring extension of $R$ contained in the normalization of $R$.Comment: Final version. The content is the same as the "Online First" version published on the journal's web sit

Computing the ideal class monoid of an order

There are well known algorithms to compute the class group of the maximal order $\mathcal{O}_K$ of a number field $K$ and the group of invertible ideal classes of a non-maximal order $R$. In this paper we explain how to compute also the isomorphism classes of non-invertible ideals of an order $R$ in a finite product of number fields $K$. In particular we also extend the above-mentioned algorithms to this more general setting. Moreover, we generalize a theorem of Latimer and MacDuffee providing a bijection between the conjugacy classes of integral matrices with given minimal and characteristic polynomials and the isomorphism classes of lattices in certain $\mathbb{Q}$-algebras, which under certain assumptions can be explicitly described in terms of ideal classes.Comment: final versio

Local isomorphism classes of fractional ideals of orders in \'etale algebras

We study the local isomorphism classes, also known as genera or weak equivalence classes, of fractional ideals of orders in \'etale algebras. We provide a classification in terms of linear algebra objects over residue fields. As a by-product, we obtain a recursive algorithm to compute representatives of the classes, which vastly outperforms previously known methods.Comment: Comments are welcom

Every finite abelian group is the group of rational points of an ordinary abelian variety over F2, F3 and F5

We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over F2, F3 and F5. We produce partial results for abelian varieties over a general finite field Fq. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over Fq when q is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over F

Abelian varieties over finite fields and their groups of rational points

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.Comment: 28 pages. Comments are welcom

Products and Polarizations of Super-Isolated Abelian Varieties

In this paper we study super-isolated abelian varieties, that is, abelian varieties over finite fields whose isogeny class contains a single isomorphism class. The goal of this paper is to (1) characterize whether a product of super-isolated varieties is super-isolated, and (2) characterize which super-isolated abelian varieties admit principal polarizations, and how many up to polarized isomorphisms.Comment: accepted versio