Related papers: Polarizations on abelian varieties

Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally…

We study the isogenies of certain abelian varieties over finite fields with non-commutative endomorphism algebras with a view to potential use in isogeny-based cryptography. In particular, we show that any two such abelian varieties with…

Let $A$ be an abelian variety defined over a number field $K$, $E/K$ be an elliptic curve, and $\phi:A\to E^m$ be an isogeny defined over $K$. Let $P\in A(K)$ be such that $\phi(P)=(Q_1,\dots, Q_m)$ with $\text{Rank}_\mathbb{Z}(\langle…

We explain the linear algebraic framework provided by Tate modules of isogenous abelian varieties in a category-theoretic way.

It is shown that to every Q-linear cycle \bar\alpha modulo numerical equivalence on an abelian variety A there is canonically associated a Q-linear cycle \alpha modulo rational equivalence on A lying above \bar\alpha. The assignment…

Let $A$ be an abelian variety of dimension $g$ together with a principal polarization $\phi: A \rightarrow \hat{A}$ defined over a field $k$. Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of…

Pareschi showed that an ample divisor of degree $d$ on a simple abelian variety of dimension $g$ has mild singularities when $d<g$. We extend his result to general polarized abelian varieties. We also show that ample divisors of degree $3$…

We prove an explicit formula for the Bergman kernel of polarized abelian varieties. As applications, we show that if two positive line bundles represent the same first Chern class and have identical Bergman kernel functions for some tensor…

The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the zeta function $L(B/K;s)$ is equivalent to the product of zeta functions of non-CM newforms for…

Assuming Lang's conjecture, we prove that for a fixed prime $p$, number field $K$, and positive integer $g$, there is an integer $r$ such that no principally polarized abelian variety $A/K$ of dimension $g$ has full level $p^r$ structure.…

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 an ordinary abelian variety $X$, $F^e_*\mathcal{O}_X$ is decomposed into line bundles for every positive integer $e$. Conversely, if a smooth projective variety $X$ satisfies this property and its Kodaira dimension is non-negative, then…

Let G be any group and F an algebraically closed field of characteristic zero. We show that any G-graded finite dimensional associative G-simple algebra over F is determined up to a G-graded isomorphism by its G-graded polynomial…

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…

Let P a locally finite partially ordered set, F a field, G a group, and I(P,F) the incidence algebra of P over F. We describe all the inequivalent elementary G-gradings on this algebra. If P is bounded, F is a infinite field of…

We present a modern proof of a theorem of Rosenlicht, asserting that every variety as in the title is isomorphic to a product of affine lines and punctured affine lines.

For every prime $p$ and integer $n\ge 3$ we explicitly construct an abelian variety $A/\F_{p^n}$ of dimension $n$ such that for a suitable prime $l$ the group of quasi-isogenies of $A/\F_{p^n}$ of $l$-power degree is canonically a dense…

For a polarized abelian variety, Z. Jiang and G. Pareschi introduce an invariant and show that the polarization is basepoint free or projectively normal if the invariant is small. Their result is generalized to higher syzygies by F. Caucci,…

In this paper, we establish a derived Torelli Theorem for twisted abelian varieties. Starting from this, we explore the relation between derived isogenies and classical isogenies. We show that two abelian varieties of dimension $\geq 2$ are…

Fix a prime number $\ell$. Graphs of isogenies of degree a power of $\ell$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a…