Isogeny classes of abelian varieties over function fields

Semisimple pointed isogeny graphs for abelian varieties

In this paper we show that if $ϕ_{i}:A_{i}\rightarrow{A}$ 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 dimension $n$ generated by $K$-rational points. We also show that the same result is true for elliptic curves without the semisimplicity condition. Furthermore, we give an explicit counterexample for abelian varieties of higher dimension to show that the semisimplicity condition is indeed necessary.

Isogeny graphs of superspecial abelian varieties

We define three different isogeny graphs of principally polarized superspecial abelian varieties, prove foundational results on them, and explain their role in number theory and geometry. This is background to joint work with Yevgeny Zaytman on properties of these isogeny graphs for dimension $g>1$, especially the result that these graphs are connected, but not in general Ramanujan.

卢宾 - 泰特型阿贝尔簇的同构类大小

Size of isogeny classes of abelian varieties of Lubin-Tate type

研究卢宾 - 泰特型阿贝尔簇同构类大小，证明了有限域上简单阿贝尔簇同构类大小的下界，并基于牛顿层预期大小猜想该下界精确。

Isogeny complexes of superspecial abelian varieties

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 abelian varieties, where the isogenies are related to quaternionic hermitian forms. We first consider isogeny graphs. We show that these $[\ell]$-isogeny graphs are a generalized Brandt graph and construct them entirely in terms of definite quaternion algebras. We prove that they are connected and give examples to show that the regular graphs obtained are someti…

Jacobians in isogeny classes of supersingular abelian threefolds in characteristic 2

We exhibit the isogeny classes of supersingular abelian threefolds over F_{2^n} containing the Jacobian of a genus 3 curve. In particular, we prove that for even n>6 there always exist a maximal and a minimal curve over F_{2^n}. All the curves can be obtained explicitly.

An ordinary abelian variety with an etale self-isogeny of p-power degree and no isotrivial factors

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 ordinary abelian varieties over function fields whose groups of points over the maximal purely inseparable extension is not finitely generated, answering in the negative a question of Thomas Scanlon.

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…

Principally polarizable isogeny classes of abelian surfaces over finite fields

Let A be an isogeny class of abelian surfaces over F_q with Weil polynomial x^4 + ax^3 + bx^2 + aqx + q^2. We show that A does not contain a surface that has a principal polarization if and only if a^2 - b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3. We use this result in a forthcoming paper in which we determine which isogeny classes of abelian surfaces over finite fields contain Jacobians.

CM-lifts for Isogeny Classes of Shimura F-crystals over Finite Fields

We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with Frobenius and other endomorphisms over $\dbF_{p^q}$, whose $p$-divisible groups in mixed characteristic $(0,p)$ are with complex multiplication.

Jacobians in isogeny classes of abelian surfaces over finite fields

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields.

Polarizations on abelian varieties

The results in this paper imply that for every number field F and positive integer r, there exists an F-isogeny class of abelian varieties such that r divides the degree of every F-polarization on every abelian variety in the isogeny class.

函数域上同源类中高度的变化

Variation of height in an isogeny class over a function field

本文研究任意特征下曲线函数域上 Abel 簇同源类中的高度变化，给出微分高度与模高度变化的最优估计，并证明平行四边形不等式及相关推论。

有限域上三维阿贝尔簇同源类中的超椭圆雅可比簇

Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields

本文针对有限域上三维阿贝尔簇同源类是否包含亏格3超椭圆曲线雅可比簇的问题，基于韦伊多项式分析提出新的阻碍判据，并猜想其中两个阻碍可在q趋于无穷时渐近分类所有同源类。

Hesse形式下的阿贝尔面与显式同源公式

Abelian surfaces in Hesse form and explicit isogeny formulas

本文提出了一种基于Hesse形式计算主极化阿贝尔面(3,3)-同源的新方法，并探讨了其在更高维和不同同源度的应用。

Isogeny graphs on superspecial abelian varieties: Eigenvalues and Connection to Bruhat-Tits buildings

We study for each fixed integer $g \ge 2$, for all primes $\ell$ and $p$ with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of $p$. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles-Goren-Lauter type cryptographic hash functio…

有限域上阿贝尔簇的Frobenius Alexander纽胚、纽着色与同构类

Frobenius Alexander quandles, knot colorings, and isogeny classes of abelian varieties over finite fields

该研究引入有限域上阿贝尔簇的Frobenius Alexander纽胚概念，确定纽被其非平凡着色的素数条件，证明其同构类可确定阿贝尔簇同构类，为纽理论与算术研究提供新工具。

Isogeny classes of abelian varieties with no principal polarizations

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 $k$ of odd prime degree $p$, and an elliptic curve $E$ over $k$ that has no complex multiplication over $k$ and that has no $k$-defined $p$-isogenies to another elliptic curve, we construct a simple $(p-1)$-dimensional abelian variety $X$ over $k$ such that every polarization of every abelian variety isogenous to $X$ has degree divisible by $p^2$. We note tha…

阿贝尔簇之间的扭曲导出等价

Twisted Derived Equivalences Between Abelian Varieties

本文研究了阿贝尔簇之间的扭曲导出等价，证明了皮卡簇的同源类是扭曲导出不变量，并给出了两个阿贝尔簇扭曲导出等价的充分必要条件。

主极化超特殊阿贝尔曲面的同源图中的顶点邻域

Neighborhood of vertices in the isogeny graph of principally polarized superspecial abelian surfaces

本研究确定了当其中一条超奇异椭圆曲线定义在$\u2119_p$上时，主极化超特殊阿贝尔曲面的$(\ell, \ell)$-同源图中对应乘积曲面顶点的局部结构，并给出了已有文献主定理的简单新证明。

On the cyclicity of the rational points group of abelian varieties over finite fields

We propose a simple criterion to know if an abelian variety $A$ defined over a finite field $\mathbb{F}_q$ is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring End$_{\mathbb{F}_q}(A)$. We also provide a criterion to know if an isogeny class is cyclic, i.e., all its varieties are cyclic; this criterion is based on the characteristic polynomial of the isogeny class. We find some asymptotic lower bounds on the fraction of cyclic $\mathbb{F}_q$-isogeny classes among certain families of them, when $q$ tends to infinity. Some of these bounds req…

计算有理数上典型主极化阿贝尔曲面的同源类

Computing isogeny classes of typical principally polarized abelian surfaces over the rationals

提出一种高效算法，计算有理数上几何自同态环为整数环的主极化阿贝尔曲面的所有同源类，并应用于1440894个亏格2曲线雅可比簇的同源类验证其有效性。

Prym varieties of étale covers of hyperelliptic curves

It is well known that the Prym variety of an étale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny is an isomorphism. We compute the degree of this isogeny in the remaining cases and show that only in the case of coverings of degree 4 it is an isomorphism.

Unlikely intersections between isogeny orbits and curves

Fix an abelian variety $A_0$ and a non-isotrivial abelian scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of translates of a fixed finite-rank subgroup of $A_0$, also defined over the algebraic numbers, by abelian subvarieties of $A_0$ of codimension at least $k$ under all isogenies between $A_0$ and some fiber of the abelian scheme. We characterize the curves inside the abelian scheme which are defined over the algebraic numbers, dominate the base curve and potentially intersect this set in infinitely many points. Our proof fol…

Isogenies of abelian Anderson A-modules and A-motives

As a generalization of Drinfeld modules, Greg Anderson introduced abelian t-modules and t-motives over a perfect field. In this article we study relative versions of these over rings. We investigate isogenies among them. Our main results state that every isogeny possesses a dual isogeny in the opposite direction, and that a morphism between abelian t-modules is an isogeny if and only if the corresponding morphism between their associated t-motives is an isogeny. We also study torsion submodules of abelian t-modules which in general are non-reduced group schemes. They can be obtained from the a…

Finite group subschemes of abelian varieties over finite fields

Let $A$ be an abelian variety over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by the Weil polynomial $f_A$. We assume that $f_A$ is separable. For a given prime number $\ell\neq\mathrm{char}\, k$ we give a classification of group schemes $B[\ell]$, where $B$ runs through the isogeny class, in terms of certain Newton polygons associated to $f_A$. As an application we classify zeta functions of Kummer surfaces over $k$.

Abelian varieties over finite fields with a specified characteristic polynomial modulo l

We estimate the fraction of isogeny classes of abelian varieties over a finite field which have a given characteristic polynomial P(T) modulo l. As an application we find the proportion of isogeny classes of abelian varieties with a rational point of order l.

主极化超特殊阿贝尔曲面之间同源的最小次数的一致界

A uniform bound for the minimal degree of an isogeny between principally polarized superspecial abelian surfaces

针对F_{p²}上具有p²-弗罗贝尼乌斯[-p]的主极化超特殊阿贝尔曲面，本文证明其间可分极化同源的乘子有一致上界p/√2，并给出达该界的多项式时间确定性算法，改进了KLPT²的相关结果。

有限域上阿贝尔簇的极化：通过典范提升

Polarizations of abelian varieties over finite fields via canonical liftings

本文通过典范提升和范畴等价，显式计算了有限域上同源类中阿贝尔簇的极化同构类，并实现了相应算法。

superspecial abelian varieties 的 isogeny 图和 Brandt 矩阵

Isogeny graphs of superspecial abelian varieties and Brandt matrices

本文研究了 superspecial abelian varieties 的 isogeny 图和 Brandt 矩阵，证明了三种 isogeny 图的连通性，并探讨了它们的性质和 Ramanujan 性质。

与超奇异阿贝尔曲面的Moret-Bailly族相关联的同源图

Isogeny graphs associated to Moret-Bailly families of supersingular abelian surfaces

研究特征p下阿贝尔曲面的Moret-Bailly族等价类与相对(ℓ,ℓ)-同源构成的有限有向图，将其与GSp₄/ℚ的内形式上的代数模形式空间关联，并利用Jacquet-Langlands对应估计邻接矩阵的特征值。

具有半稳定约化的 Motive 同构类中高度的有限性

Finiteness of Heights in Isogeny Classes of Motives with Semistable Reduction

本文利用积分 $p$-adic Hodge 理论推广了阿贝尔簇的 Faltings 高度，并在假设 adelic Mumford-Tate 猜想下，证明了 motive 同构类中高度的有限性，扩展了 Kisin 和 Mocz 的相关结果。

亏格3曲线的一些显式算术及其应用

Some explicit arithmetic on curves of genus three and their applications

本文发展了亏格3曲线的显式算术，提出了计算分解Richelot同源余域的算法及Howe曲线定义方程的显式公式，并据此构建了枚举超特殊Howe曲线的高效算法。

通过 pro-étale 基本群研究超奇异阿贝尔曲面的拟同源群

Quasi-isogeny groups of supersingular abelian surfaces via pro-étale fundamental groups

本文通过 pro-étale 基本群分类非正规概型的几何覆盖并计算曲线基本群，证明了 Siegel 三重 fold 上的 torsor 诱导出超奇异阿贝尔曲面自拟同源群与有限生成自由群的同构。

On classification of groups of points on abelian varieties over finite fields

In this paper we improve our previous results on classification of groups of points on abelian varieties over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given $k$-isogeny class.

Counting Richelot isogenies between superspecial abelian surfaces

Castryck, Decru, and Smith used superspecial genus-2 curves and their Richelot isogeny graph for basing genus-2 isogeny cryptography, and recently, Costello and Smith devised an improved isogeny path-finding algorithm in the genus-2 setting. In order to establish a firm ground for the cryptographic construction and analysis, we give a new characterization of {\em decomposed Richelot isogenies} in terms of {\em involutive reduced automorphisms} of genus-2 curves over a finite field, and explicitly count such decomposed (and non-decomposed) Richelot isogenies between {\em superspecial} principal…

Families of abelian varieties with many isogenous fibres

Let Z be a subvariety of the moduli space of principally polarised abelian varieties of dimension g over the complex numbers. Suppose that Z contains a Zariski dense set of points which correspond to abelian varieties from a single isogeny class. A generalisation of a conjecture of André and Pink predicts that Z is a weakly special subvariety. We prove this when dim Z = 1 using the Pila--Zannier method and the Masser--Wüstholz isogeny theorem. This generalises results of Edixhoven and Yafaev when the Hecke orbit consists of CM points and of Pink when it consists of Galois generic points.

有限域上阿贝尔簇乘积的主极化

Principal polarizations on products of abelian varieties over finite fields

本文改进并推广有限域上阿贝尔簇乘积主极化的已有结果，分析通常与超奇异阿贝尔簇乘积同源类中不可约主极化缺失的原因，并给出几何单阿贝尔曲面与椭圆曲线情形的存在条件，证明相应阿贝尔三维簇或其二次扭曲为 Jacobian。

Commutative algebraic groups up to isogeny

Consider the abelian category $\mathcal{C}_k$ of commutative group schemes of finite type over a field $k$. By results of Serre and Oort, $\mathcal{C}_k$ has homological dimension $1$ (resp. $2$) if $k$ is algebraically closed of characteristic $0$ (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of $\mathcal{C}_k$ by the full subcategory $\mathcal{F}_k$ of finite $k$-group schemes. We show that $\mathcal{C}_k/\mathcal{F}_k$ has homological dimension $1$, and we determine its projective or injective object…

Spanning the isogeny class of a power of an ordinary elliptic curve over a finite field. Application to the number of rational points of curves of genus $\leq 4$

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 of $E^g$. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null…

On the Tate and Langlands--Rapoport conjectures for special fibres of integral canonical models of Shimura varieties of abelian type

We prove the isogeny property for special fibres of integral canonical models of compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type. The approach used also shows that many crystalline cycles on abelian varieties over finite fields which are specializations of Hodge cycles, are algebraic. These two results have many applications. First, we prove a variant of the conditional Langlands--Rapoport conjecture for these special fibres. Second, for certain isogeny sets we prove a variant of the unconditional Langlands--Rapoport conjecture (like for many basic loci). Third, we prov…

On the Structure of the Weil Restriction of Abelian Varieties

We give a description of endomorphism rings of Weil restrictions of abelian varieties with respect to finite Galois extensions. The results are applied to study the isogeny decomposition of Weil restrictions.

有限域上的不含曲率3或更小的阿贝尔表面

Abelian surfaces over finite fields containing no curves of genus $3$ or less

本文研究了有限域上不含曲率3或更小曲线的阿贝尔表面，通过极化度与曲率的关系，完善了同源类的特征化，并描述了相关曲线的性质。

Weil多项式形式为$t^{2g}+at^g+q^g$的某些算术性质

Some arithmetic properties of Weil polynomials of the form $t^{2g}+at^g+q^g$

本文研究了Weil多项式形式为$t^{2g}+at^g+q^g$的同源类的局部循环性及有理点群的局部增长。

关于函数域上阿贝尔簇的$\mu$-不变量和同源

On $μ$-invariants and isogenies for abelian varieties over function fields

本文给出了函数域上阿贝尔簇在非分歧$\mathbb{Z}_{p}$扩张中Iwasawa $\mu$-不变量在同源下的变化公式，并证明了Birch--Swinnerton-Dyer猜想在同源下的不变性。

Abelian varieties with quaternion and complex multiplication

In this paper we study abelian varieties which correspond to CM points in the coarse moduli space of principally polarized abelian varieties with multiplication by a maximal order in a quaternion algebra over a totally real number field. These are abelian varieties of even dimension with quaternion and complex multiplication. We describe them explicitly via isogenies to products of abelian varieties of smaller dimension together with estimates on the degree of the isogeny.

Abelian varieties with finite abelian group action

An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan's decomposition theorem. We show that they are essentially the same. Moreover, we generalize in a sense this result to abelian varieties with action of an arbitrary finite abelian group.

阿贝尔簇上零维Chow群的滤过及其在同源下的行为

Filtrations of the Chow group of zero-cycles on abelian varieties and behavior under isogeny

本文研究阿贝尔簇零维Chow群上的Bloch-Beilinson型滤过在同源下的行为，证明乘以n映射在商群上诱导乘以n^r，并在椭圆曲线乘积情形与已知滤过一致，同时为Raskind-Spiess和Colliot-Thélène猜想提供新证据。

复阿贝尔簇与量子纠错：GKP码的数学框架

Complex abelian varieties and quantum error correction: a mathematical framework for GKP codes

本文通过复阿贝尔簇的几何研究GKP量子纠错码，建立其与阿贝尔簇理论的精确对应关系，并证明编码渐近等距、逻辑Clifford门由高斯幺正实现等关键结果。

Products and Polarizations of Super-Isolated Abelian Varieties

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 super-isolated varieties is super-isolated, and (2) characterize which super-isolated abelian varieties admit principal polarizations, and how many up to polarized isomorphisms.

GV-subschemes and their embeddings in principally polarized abelian varieties

We prove that the embedding of a $GV$-subscheme in a principally polarized abelian variety does not factor through any nontrivial isogeny. As an application, we present a new proof of a theorem of Clemens--Griffiths identifying the intermediate Jacobian of a smooth cubic threefold in $\mathbf{P}^4$ to the Albanese variety of its Fano surface of lines.

The Severi problem for abelian surfaces in the primitive case

We prove that the irreducible components of primitive class Severi varieties of general abelian surfaces are completely determined by the maximal factorization through an isogeny of the maps from the normalized curves.

例外Shimura品种上函数域值点的有限性

Finiteness of function field-valued points on exceptional Shimura varieties

研究证明了例外Shimura品种上函数域值点的有限性，并推广到更一般的Shimura品种和素数条件。

Decomposed Richelot isogenies of Jacobian varieties of curves of genus 3

For a nonsingular projective curve $C$ of genus 3 defined over an algebraically closed field of characteristic $p > 2$, we give a necessary and sufficient condition that the Jacobian variety $J(C)$ has a decomposed Richelot isogeny outgoing from it and we determine the structures of decomposed ones.

Albanese varieties of abelian covers

We show that Albanese varieties of abelian covers of projective plane are isogenous to product of isogeny components of abelian varieties associated with singularities of the ramification locus. In particular Albanese varieties of abelian covers of projective plane ramified over arrangements of lines and uniformized by unit ball are isogenous to a product of Jacobians of Fermat curves. Periodicity of the sequence of (semi-abelian) Albanese varieties of unramified cyclic covers of complements to a plane singular curve is shown.