Jacobians in isogeny classes of abelian surfaces over finite fields
Everett W. Howe, Enric Nart, Christophe Ritzenthaler
对于有限域上亏格为2的曲线，我们给出了一个完整的答案，即哪些多项式是Frobenius的特征多项式。
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.
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Jacobians in isogeny classes of supersingular abelian threefolds in characteristic 2
给出了包含亏格为3的曲线的Jacobian的F_{2^n}上超奇异三重交换的等距类.特别地，我们证明了即使n>6，在F_{2^n}上总是存在一条极大曲线和一条极小曲线。所有的曲线都可以显式地得到。
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。
A Note on Jacobians with Prescribed Factors
本文在无限域上改进了Matsusaka定理，使得补同源因子可以是任意大维数的绝对单因子，并讨论了相对单性与同源Schottky问题的相关问题。
We give parametrisation of curves C of genus 2 with a maximal isotropic (ZZ/3)^2 in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it can shown that non-reducible Jacobians have nontrivial 3-part of the Tate-Shafarevich group.
We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.
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。
A framework for Tate modules of abelian varieties under isogeny
本文基于同源映射下的阿贝尔簇的泰特模线性代数框架，从范畴论角度探讨其结构与性质。
We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of $\mathcal{D}$-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.
Principally polarizable isogeny classes of abelian surfaces over finite fields
设A是F_q上具有Weil多项式x^4 + ax^3 + bx^2 + aqx + q^2的交换曲面的等距类。我们证明了A不包含具有主极化的曲面当且仅当a^2 - b = q且b < 0且b的所有素除子全等于1模3。我们使用这一结果在即将发表的论文中，我们确定哪些isobian类的阿贝尔曲面在有限域包含雅可比。
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.
Abelian varieties with real multiplication: classification and isogeny classes over finite fields
本文在关于Newton多边形的温和假设下分类了有限域上Hilbert模簇的特定点，并据此证明了同构类大小的估计。
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…
On maps between modular Jacobians and Jacobians of Shimura curves
利用Jacquet-Langlands对应和Faltings同构定理，当p和q是不整除D或N的不同素数时，阿贝尔簇J_0^{Dpq}（N）（判别式Dpq的具有Gamma_0（N）水平结构的Shimura曲线的Jacobian）与J_0 ^D（pqN）的pq-新子簇同构.然而，这种方法是完全非建设性的，并没有提供任何信息，在所有关于任何特定的两个品种之间的差异。本文确定了Hecke代数的一个有限小的极大理想集上从一个簇到另一个簇的所有可能映射的核。我们还研究了这种计算的后果的几何形状的Jacobians志村曲线。特别是，我们得到了“重一”的结果，某些伽罗瓦表示的挠的雅可比曲线。
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。
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…
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。
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.
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…
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。
Polarizations on abelian varieties
本文的结果表明：对任意数域F和正整数r，存在一个阿贝尔簇的F-等距类，使得r整除等距类中任意阿贝尔簇上的F-极化度.
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。
In this paper, we study the Cassels-Tate pairing on Jacobians of genus two curves admitting a special type of isogenies called Richelot isogenies. Let $ϕ: J \rightarrow \widehat{J}$ be a Richelot isogeny between two Jacobians of genus two curves. We give an explicit formula as well as a practical algorithm to compute the Cassels-Tate pairing on $\text{Sel}^{\widehatϕ}(\widehat{J}) \times \text{Sel}^{\widehatϕ}(\widehat{J})$ where $\widehatϕ$ is the dual isogeny of $ϕ$. The formula and algorithm are under the simplifying assumption that all two torsion points on $J$ are defined over $K$. We als…
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…
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…
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…
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 of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also …
Spanning isogeny classes of principally polarized abelian surfaces with RM
本文提出将具有实乘的主极化阿贝尔曲面间的同构分解为基本类型，从而提供枚举给定PPAS同构类的策略。
This document is intended to summarize the theory and methods behind fq_isog collection inside the ab_var database in the LMFDB as well as some observations gleaned from these databases. This collection consists of tables of Weil q-polynomials, which by the Honda-Tate theorem are in bijection with isogeny classes of abelian varieties over finite fields.
Let $\frak{p}$ and $\frak{q}$ be two distinct prime ideals of $\mathbb{F}_q[T]$. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve $X_0(\frak{p}\frak{q})$ to compare the rational torsion subgroup of the Jacobian $J_0(\frak{p}\frak{q})$ with its subgroup generated by the cuspidal divisors, and to produce explicit examples of Jacquet-Langlands isogenies. Our results are stronger than what is currently known about the analogues of these problems over $\mathbb{Q}$.
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。
We construct six infinite series of families of pairs of curves (X,Y) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3, or 4. For each family, we compute the isomorphism type of the isogeny kernel and the dimension of the image of the family in the appropriate moduli space. The families are derived from Cassou--Noguès and Couveignes' explicit classification of pairs (f,g) of polynomials such that f(x_1) - g(x_2) is reducible.
Ordinary abelian varieties: isogeny graphs and polarizations
本文提出计算普通阿贝尔簇中度数整除D的同源和极化的算法，并探讨了高维阿贝尔簇中特有的现象及同源图的分解方法。
Given a polynomial $f\in\mathbb{C}[x]$, we consider the family of superelliptic curves $y^d=f(x)$ and their Jacobians $J_d$ for varying integers $d$. We show that for any integer $g$ the number of abelian varieties up to isogeny of dimension $\le g$ which appear in any $J_d$ is finite and their multiplicities are bounded.
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。
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 same as the logarithmic asymptotic of the number $m_q(g)$ of isogeny classes of all abelian varieties of dimension $g$ over $\mathbb{F}_q$. We also prove that $$ \limsup_{g \rightarrow \infty} \frac{s_q(g)}{m_q(g)}=1. $$ This suggests that there are much more simple isogeny classes of abelian varieties over $\mathbb{F}_q$ of dimension $g$ than non-simple ones f…
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。
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。
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.
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。
Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed ${g\geq1}$, the proportion of isogeny classes of $g$ dimensional abelian varieties defined over $\mathbb{F}_q$ which fail this condition is $1 - Q(2g + 2)/2^g$ as $q\to\infty$ ranges over odd prime powers, where $Q(n)$ denotes the number of partitions of…
Abelian varieties over finite fields with a specified characteristic polynomial modulo l
我们估计了有限域上具有给定模l特征多项式P（T）的阿贝尔簇的同构类的分数。作为应用，我们求出了具有l阶有理点的阿贝尔簇的等距类的比例.
Isogeny graphs of abelian varieties and singular ideals in orders
本文推广Kohel的火山图定理至任意维阿贝尔簇，通过理想论方法研究含局部Bass序的同源图结构，适用于非单非普通情形，并给出结构定理与意外例子。
An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a centr…
Let $p$ and $q$ be distinct primes, and let $X_{p,q}$ be the $(q+1)$-regular graph whose nodes are supersingular elliptic curves over $\overline{\mathbb{F}}_p$ and whose edges are $q$-isogenies. For fixed $p$, we compute the distribution of the $\ell$-Sylow subgroup of the sandpile group (i.e.\ Jacobian) of $X_{p,q}$ as $q \to \infty$. We find that the distribution disagrees with the Cohen-Lenstra heuristic in this context. Our proof is via Galois representations attached to modular curves. As a corollary of our result, we give an upper bound on the probability that the Jacobian is cyclic, whi…
Isogeny classes of non-simple abelian surfaces over finite fields
本文研究有限域上主极化几乎普通分裂阿贝尔曲面的同源类计数问题，给出了与给定曲面同源的主极化阿贝尔曲面数量的渐近上下界，细化了 Achter 和 Howe 的已有结果。
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。
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 that certain questions about polarised isogenies can be reduced to questions about unpolarised isogenies or vice versa. Our main theorem concerns abelian varieties B which are isogenous to a fixed abelian variety A. It establishes the existence of a polarised isogeny A to B whose degree is polynomially bounded in n, if there exist both an unpolarised isogeny…
On Weil Polynomials of Hyperelliptic Curves over Finite Fields of Characteristic 2
本文通过模2剩余类约束给出特征2有限域上超椭圆雅可比矩阵存在性的新阻碍条件，证明Weil系数奇偶性渐近等分布，并据此排除大量3维和4维同源类，同时提出运行时间显著改进的任意亏格超椭圆曲线枚举算法。
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。
For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-3 isogeny on $A$ factors as a composition of 3-isogenies over $F$. This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension one, we deduce that if $E/F$ is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded.…
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…
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.
Let $J^{65}$ be the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $65$. We study the isogenies $J_0(65)\rightarrow J^{65}$ defined over $\mathbb{Q}$, whose existence was proved by Ribet. We prove that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on $J_0(65)$, and moreover the odd part of the kernel is generated by a cuspidal divisor of order $7$, as is predicted by a conjecture of Ogg.
Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multiplication. Any subgroup of the class group, and generating set thereof, induces an isogeny graph on the orbit of the variety for this subgroup. We compute (under the Generalized Riemann Hypothesis) some bounds on the norms of prime ideals generating it, such that the associated graph has good expansion properties. We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and Robert …
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 finite field. We analyse graphs of so-called $\mathfrak l$-isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as $(\ell, \ell)$-isogenies: those whose kernels are maximal isotropi…
We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.
Weil numbers generated by other Weil numbers and torsion fields of abelian varieties
利用性质的Frobenius特征值，我们表明，在一个精确的意义上，“最”同构类（主要是极化）简单的阿贝尔簇在有限域的特点，直到issues由其序列的划分领域，和一个类似的结果为“最”issues类。一些全球病例也得到治疗。
Abelian surfaces of GL2-type as Jacobians of curves
本文研究了GL 2型阿贝尔簇的主极化同构类集。作为结果的应用，我们构造了\bar上的亏格为2的曲线C，C '/\Q，它们在\Q上是非同构的，而在\Q上共享同构的非极化模雅可比簇;我们还给出了一种方法，以获得\Q上亏格为2的曲线，其雅可比簇同构于二次\Q -曲线的Weil限制，并给出了例子.
For a complex abelian variety $A$ with endomorphism ring isomorphic to the maximal order in a quartic CM-field $K$, the Igusa invariants $j_1(A), j_2(A),j_3(A)$ generate an abelian extension of the reflex field of $K$. In this paper we give an explicit description of the Galois action of the class group of this reflex field on $j_1(A),j_2(A),j_3(A)$. We give a geometric description which can be expressed by maps between various Siegel modular varieties. We can explicitly compute this action for ideals of small norm, and this allows us to improve the CRT method for computing Igusa class polynom…
Small degree isogenies between conjugate principally polarized superspecial abelian surfaces
本文证明在有限域上每个主极化超特殊阿贝尔曲面到其Frobenius共轭存在小乘子同源，推广了超奇异椭圆曲线的已知结果。
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