相关论文: Polarizations on abelian varieties

Let $A$ be an abelian variety of Mumford's type. This paper determines all possible Newton polygons of $A$ in char $p$. This work generalizes a result of R. Noot.

In this article, we prove that every finite abelian group $G$ of odd order occurs as a subgroup of the class group of infinitely many real cyclotomic fields.

We prove an inclusion result for graded dagger closure for primary ideals in symmetric section rings of abelian varieties over an algebraically closed field of arbitrary characteristic.

By the algebraization of affine Nash groups, a connected affine Nash group is an abelian Nash manifold if and only if its algebraization is a real abelian variety. We first classify real abelian varieties up to isomorphisms. Then with a bit…

We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group $G$, the sequence of the numbers $E(G,m)$…

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 show that any supersingular abelian variety is isogenous to a superspecial abelian variety without increasing field extensions. The proof uses minimal isogenies and the Galois descent. We then construct a superspecial…

We give a formula for the class number of an arbitrary CM algebraic torus over $\mathbb{Q}$. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected…

In this paper, we investigate the asymptotic behavior of the number $s_q(g)$ of isogeny classes of simple abelian varieties of dimension $g$ over a finite field $\mathbb{F}_q$. We prove that the logarithmic asymptotic of $s_q(g)$ is the…

We classify, up to equivalence, all finite-dimensional simple graded division algebras over the field of real numbers. The grading group is any finite abelian group.

Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class…

We give a presentation of abelian class field theory.

Let A be the moduli space of principally polarized abelian varieties of dimension 4 over an algebraically closed field k of characteristic different from 2,3. It is proved that the universal principally polarized abelian variety over A, as…

We study the homogeneous involutions on the full square matrices over an algebraically closed field endowed with a division grading with commutative support. We obtain the classification of the isomorphism and equivalence classes for the…

Let $F$ be a totally real field with ring of integers $O_F$, and $D$ be a totally definite quaternion algebra over $F$. A well-known formula established by Eichler and then extended by K\"orner computes the class number of any $O_F$-order…

We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to…

If A and B are abelian varieties over a number field K such that there are non-trivial geometric homomorphisms of abelian varieties between reductions of A and B at most primes of K, then there exists a non-trivial (geometric) homomorphism…

In this paper we prove a general theorem concerning the number of translation classes of curves of genus $g$ belonging to a fixed cohomology class in a polarized abelian variety of dimension $g$. For $g = 2$ we recover results of G\"ottsche…

Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Neron model of A/K has at least one fiber with potential toric dimension one, then for almost all rational primes…

We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite…