Related papers: Ordinary abelian varieties: isogeny graphs and pol…

We call an abelian variety over a finite field $\mathbb{F}_q$ super-isolated if its ($\mathbb{F}_q$-rational) isogeny class contains a single isomorphism class. In this paper, we use the Honda-Tate theorem to characterize super-isolated…

We consider the distribution of p-power group schemes among the torsion of abelian varieties over finite fields of characteristic p, as follows. Fix natural numbers g and n, and let $\xi$ be a non-supersingular principally quasipolarized…

We discuss various constructions which allow one to embed a principally polarized abelian variety in the jacobian of a curve. Each of these gives representatives of multiples of the minimal cohomology class for curves which in turn produce…

We determine the set of geometric endomorphism algebras of geometrically split abelian surfaces defined over $\mathbb{Q}$. In particular we find that this set has cardinality 92. The essential part of the classification consists in…

We consider three isogeny invariants of abelian varieties over finite fields: the Galois group, Newton polygon, and the angle rank. Motivated by work of Dupuy, Kedlaya, and Zureick-Brown, we define a new invariant called the weighted…

It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is…

Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by…

It is proved that the universal degree bound for separating polynomial invariants of a finite abelian group (in non-modular characteristic) is strictly smaller than the universal degree bound for generators of polynomial invariants, unless…

We prove that the automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields are cyclic, and give a complete list of finite groups that can be realized as such automorphism groups.

In this paper we show that if $\phi_{i}:A_{i}\rightarrow{A}$ is a semisimple pointed $K$-rational $\ell$-isogeny graph of order $n$ for a prime $\ell$, then the group of $\ell$-torsion points $A[\ell](\overline{K})$ contains a subspace of…

We establish correspondances between factorisations of finite abelian groups (direct factors, unitary factors, non isomorphic subgroup classes) and factorisations of integer matrices. We then study counting functions associated to these…

For two supersingular elliptic curves $E$ and $E'$ defined over $\mathbb{F}_{p^2}$, let $[E \times E']$ be the superspecial abelian surface with the principal polarization $\{0\} \times E' + E \times \{0\}$. We determine local structure of…

We present new criteria that obstruct an isogeny class of abelian varieties over a finite field with a given Weil polynomial from containing a Jacobian of a genus-3 hyperelliptic curve. Based on our analysis of the Weil polynomials of…

The endomorphism ring End(A) of an abelian variety A is an order in a semi-simple algebra over Q. The co-index of End(A) is the index to a maximal order containing it. We show that for abelian varieties of fixed dimension over any…

Let $\mathbb F$ be an algebraically closed field, $G$ be an abelian group, and let $A$ and $B$ be arbitrary finite-dimensional $G$-graded simple algebras over $\mathbb F$. We prove that $A$ and $B$ are isomorphic if, and only if, they…

An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same…

In this article we study the endomorphism algebras of abelian varieties $A$ defined over a given number field $K$ with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of $A$ to be…

In this paper, we investigate Weil polynomials and their relationship with isogeny classes of abelian varieties over finite fields. We give a necessary condition for a degree 12 polynomial with integer coefficients to be a Weil polynomial.…

We construct, for every prime p, a function field K of characteristic p and an ordinary abelian variety A over K, with no isotrivial factors, that admits an etale self-isogeny of p-power degree. As a consequence, we deduce that there exist…

For a digraph $\Gamma$, if $F$ is the smallest field that contains all roots of the characteristic polynomial of the adjacency matrix of $\Gamma$, then $F$ is called the splitting field of $\Gamma$. The extension degree of $F$ over the…