Polarized Pushouts for Finite Fields

Automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields

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.

Smooth quotients of principally polarized abelian varieties

We give an explicit characterization of all principally polarized abelian varieties $(A,Θ)$ such that there is a finite subgroup of automorphisms $G$ of $A$ that preserve the numerical class of $Θ$, and such that the quotient variety $A/G$ is smooth. We also give a complete classification of smooth quotients of Jacobians of curves.

A classification of the automorphism groups of polarized abelian threefolds over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the automorphism groups of polarized abelian threefolds over finite fields.

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.

Absolutely simple polarized abelian varieties of odd Sophie Germain prime dimension over finite fields with maximal automorphism groups

For each Sophie Germain prime $g \geq 5,$ we construct an absolutely simple polarized abelian variety of dimension $g$ over a finite field, whose automorphism group is a cyclic group of order $4g+2$. We also provide a description on the asymptotic behavior of prime numbers that are related to our construction.

Variations in the distribution of principally polarized abelian varieties among isogeny classes

We show that for a large class of rings $R$, the number of principally polarized abelian varieties over a finite field in a given simple ordinary isogeny class and with endomorphism ring $R$ is equal either to 0, or to a ratio of class numbers associated to $R$, up to some small computable factors. This class of rings includes the maximal order of the CM field $K$ associated to the isogeny class (for which the result was already known), as well as the order $R$ generated over $\mathbf{Z}$ by Frobenius and Verschiebung. For this latter order, we can use results of Louboutin to estimate the appr…

有限域上主极化阿贝尔簇的上同调算术统计

Cohomological Arithmetic Statistics for Principally Polarized Abelian Varieties over Finite Fields

本文研究有限域上主极化阿贝尔簇有理点计数的分布规律，通过计算低维情形的上同调与欧拉示性数并借助上同调稳定性，提出高维极限下点计数渐近行为的猜想。

Normal projective varieties admitting polarized or int-amplified endomorphisms

Let $X$ be a normal projective variety admitting a polarized or int-amplified endomorphism $f$. We list up characteristic properties of such an endomorphism and classify such a variety from the aspects of its singularity, anti-canonical divisor and Kodaira dimension. Then we run the equivariant minimal model program with respect to not just the single $f$ but also the monoid $SEnd(X)$ of all surjective endomorphisms of $X$, up to finite-index. Several applications are given. We also give both algebraic and geometric characterizations of toric varieties via polarized endomorphisms.

On a cohomological generalization of the Shafarevich conjecture for K3 surfaces

The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. André proved this conjecture for polarized K3 surfaces of fixed degree, and recently She proved it for polarized K3 surfaces of unspecified degree. In this paper, we prove a certain generalization of their results, which is stated by the unramifiedness of l-adic étale cohomology groups for K3 surfaces over finitely generated fields of characteristic 0. As a corollary, we get the original Shafare…

二维有限-gap施罗德inger算子及其谱允许两个反演

2-dimensional finite-gap Schrödinger operator whose spectrum admits two involutions

本文研究了二维有限-gap施罗德inger算子的势条件，并提出了一种识别特定类曲线双覆盖的isoPrymians的方法。