Related papers: A note on supersingular abelian varieties

We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.

We provide a simple method of constructing isogeny classes of abelian varieties over certain fields $k$ such that no variety in the isogeny class has a principal polarization. In particular, given a field $k$, a Galois extension $\ell$ of…

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of…

An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of…

We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its…

We prove that if a curve of a non-isotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety then it is special.

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over…

Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\rm End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties over $k$…

We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford-Tate group. We prove that supersingular abelian surfaces, respectively threefolds, can be lifted to ones…

We discuss the notion of polarised isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarisations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show…

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…

We consider the structures formed by isogenies of abelian varieties with polarizations that are not necessarily principal, specifically with the $[\ell]$-polarizations we have previously defined. Our primary interest is in superspecial…

We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose…

A celebrated theorem of Bogomolov asserts that the $\ell$-adic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic $p$: a…

We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties…

Given an integer $D$ and an ordinary isogeny class of abelian varieties defined over a finite field $\mathbb{F}_q$ with commutative $\mathbb{F}_q$-endomorphism algebra, we provide algorithms for computing all isogenies of degree dividing…

In this paper we give conditions under which two abelian varieties that are defined over a finite field $F$, and are isogenous over some larger field, are $F$-isogenous. Further, we give conditions under which a given isogeny is defined…

Let $K$ be a field, $L$ a finite Galois extension of $K$, and $X$ an abelian variety defined over $L$. If $X$ is isogenous over $L$ to an abelian variety defined over $K$, then the $\ell$-adic Galois representations associated to $X$ extend…

In this Note we present an expository account for \dieu modules and revisit supersingular abelian varieties. We give a simple proof of the uniqueness of products of two or more supersingular elliptic curves (a theorem due to Deligne, Ogus…

In this note we show that if an abelian variety possesses a Galois embedding into some projective space, then it must be isogenous to the self product of an elliptic curve. We prove moreover that the self product of an elliptic curve always…