Mass formula for supersingular abelian surfaces

Fully maximal and fully minimal abelian varieties

We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus $3$ in c…

On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction

We give a short proof of the "prime-to-$p$ version" of the Manin-Mumford conjecture for an abelian variety over a number field, when it has supersingular reduction at a prime dividing $p$, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither $p$-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the prime-to-$p$ torsion points via nontrivial homothety.

超奇异阿贝尔簇在维数 $g\leq 5$ 上的模空间

A Computational Framework for the Moduli of Supersingular Abelian Varieties

建立极化旗型商的结构定理，证明顶层精细正规形和拟极化的可计算下降准则，完成旗型第一和最后一步分类，并计算 $g\leq 5$ 维的极化旗型商。

Supersingular reduction of Kummer surfaces in residue characteristic $2$

Given an abelian surface $A$, defined over a discrete valuation field and having good reduction, does the attached Kummer surface $\mathrm{Km}(A)$ also have good reduction? In this paper we give an affirmative answer in the extreme case, that is, when the abelian surface has supersingular reduction in characteristic $2$.

Dieudonné模块与奇超椭圆曲线的再审视介绍

Introduction to Dieudonné modules and supersingular abelian varieties revisited

本文介绍了Dieudonné模块和奇超阿贝尔簇，给出了奇超椭圆曲线乘积唯一性和Oort定理的简单证明。

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。

非超奇异情形下广义Kummer曲面的好约化

The good reduction of generalized Kummer surfaces in the non-supersingular case

本文针对非超奇异情形，建立了阿贝尔曲面带循环群作用时广义Kummer曲面好约化的判别准则，推广了前人关于Kummer曲面的相关结果。

Abelian surfaces with supersingular good reduction and non semisimple Tate module

We show the existence of abelian surfaces $A$ over $\mathbb{Q}_p$ having good reduction with supersingular special fibre whose associated $p$-adic Galois module $V_p(A)$ is not semisimple.

Formal groups, supersingular abelian varieties and tame ramification

Let us consider an abelian variety defined over $\mathbb{Q_{\ell}}$ with good supersingular reduction. In this paper we give explicit conditions that ensure that the action of the wild inertia group on the $\ell$-torsion points of the variety is trivial. Furthermore we give a family of curves of genus 2 such that their Jacobian surfaces have good supersingular reduction and satisfy these conditions. We address this question by means of a detailed study of the formal group law attached to abelian varieties.

超奇异阿贝尔曲面在 $\mathbb{Q}_p$ 上的单值群

Monodromy Groups of Supersingular Abelian Surfaces over $\mathbb{Q}_p$

本文对素数p≥7，参数化了Q_p上超奇异好约化阿贝尔曲面的p-adic Tate模的过滤φ-模，并由此确定了相关p-adic表示的单值群的单位分支（直到Q_p-同构），分析了其在模空间中的p-adic分布，证明单位分支一般同构于GL_2 ×_det GL_2。

A note on supersingular abelian varieties

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 abelian variety which not directly defined over a finite field. This answers negatively to a question of the author [J. Pure Appl. Alg., 2013] concerning of endomorphism algebras occurring in Shimura curves. Endomorphism algebras of supersingular elliptic curves over an arbitrary field are also investigated. We correct a main result of the author's paper [Math. Re…

Functional equations for supersingular abelian varieties over $\mathbf{Z}_p^2$-extensions

Let $K$ be an imaginary quadratic field and $K_\infty$ be the $\mathbf{Z}_p^2$-extension of $K$. Answering a question of Ahmed and Lim, we show that the Pontryagin dual of the Selmer group associated to a supersingular polarized abelian variety admits an algebraic functional equation. The proof uses the theory of $Γ$-system developed by Lai, Longhi, Tan and Trihan. We also show the algebraic functional equation holds for Sprung's chromatic Selmer groups of supersingular elliptic curves along $K_\infty$.

The supersingular locus in Siegel modular varieties with Iwahori level structure

We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case $g$ is even.

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。

超奇异性与超特殊性的阿贝尔曲面验证

Supersingularity and Superspeciality Verification of Abelian Surfaces

本文提出高效算法验证阿贝尔曲面的超奇异性和超特殊性，基于结构分析实现对数时间复杂度，并推广到任意维数。

On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions

Let $F$ be a number field unramified at an odd prime $p$ and $F_\infty$ be the $\mathbf{Z}_p$-cyclotomic extension of $F$. Let $A$ be an abelian variety defined over $F$ with good supersingular reduction at all primes of $F$ above $p$. Büyükboduk and the first named author have defined modified Selmer groups associated to $A$ over $F_\infty$. Assuming that the Pontryagin dual of these Selmer groups are torsion $\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-modules, we give an explicit sufficient condition for the rank of the Mordell-Weil group $A(F_n)$ to be bounded as $n$ varies.

某些芒福德阿贝尔四维簇的无穷多个超奇异素数

Infinitely many supersingular primes for some Mumford's abelian fourfolds

本文将 Elkies 关于实数域上椭圆曲线超奇异素数无穷性的结果推广到 Mumford 族中的某些阿贝尔四维簇及更一般的 Kuga-Satake 阿贝尔簇族，通过研究 Hodge 型 Shimura 簇的局部形变空间和 Shimura 曲线的实点完成证明。

Group structures of elementary supersingular abelian varieties over finite fields

Let A be a supersingular abelian variety over a finite field k. We give an approximate description of the structure of the group A(k) of rational points of A over k in terms of the characteristic polynomial f of the Frobenius endomorphism of A relative to k. If f=g^e for a monic irreducible polynomial g and a positive integer e, we show that there is a group homomorphism A(k) --> (Z/g(1)Z)^e whose kernel and cokernel are elementary abelian 2-groups. In particular, this map is an isomorphism if the characteristic of k is 2 or A is simple of dimension greater than 2; in the last case one has e=1…

Supersingular abelian surfaces and Eichler class number formula

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 Körner computes the class number of any $O_F$-order in $D$. In this paper we generalize the Eichler class number formula so that it works for arbitrary $\mathbb{Z}$-orders in $D$. The motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a prime finite field $\mathbb{F}_p$. We give explicit formulas for the number of these isomorphism classes for all prim…

奇特征下超奇异阿贝尔簇的Oort猜想

Oort's conjecture on supersingular abelian varieties in odd characteristic

针对奇特征p>2，构造含a不变量g-2开稠密集的闭子簇，为Oort猜想提供独立证明，补充了Viehmann对a=1轨迹的研究。

On the weak Leopoldt conjecture and coranks of Selmer groups of supersingular abelian varieties in $p$-adic Lie extensions

Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not neccesasily containing the cyclotomic $\Zp$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that…

Kummer面在非超奇异情况下的模2约简

Reduction of Kummer surfaces modulo 2 in the non-supersingular case

研究Kummer面在非超奇异情况下模2的良约简条件，构造了代数空间模型和方案模型的等价性。

Supersingular curves of genus four in characteristic two

We describe the intersection of the Torelli locus $j(\mathcal{M}_4^{ct}) = \mathcal{J}_4$ with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic two. We show that the locus of supersingular Jacobians $\mathcal{S}_4\cap\mathcal{J}_4$ in characteristic two is pure of dimension three. One way to obtain that result uses an analysis of the data of smooth genus four curves and principally polarized abelian fourfolds defined over $\mathbb{F}_2$, and another involves studying the Ekedahl-Oort types and using the indecomposability of some $\mathrm{BT}_1$ group schemes.

通过Shimura--Taniyama方法的超奇异曲线

Supersingular curves via the Shimura--Taniyama method

本文研究具有三分枝阿贝尔覆盖的曲线在正特征下为超奇异的条件，利用Shimura-Taniyama方法对亏格至多10的曲线给出完整分类，并发现相关素数密度高于预期。

Explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and applications

Rank-2 Drinfeld modules are a function-field analogue of elliptic curves, and the purpose of this paper is to investigate similarities and differences between rank-2 Drinfeld modules and elliptic curves in terms of supersingularity. Specifically, we provide an explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and prove several basic properties. As an application, we give a numerical example of an asymptotically optimal tower of Drinfeld modular curves.

阿贝尔曲面的超奇异素数分布

Distribution of supersingular primes for abelian surfaces

本文针对数域上的绝对单阿贝尔曲面，依据自同态代数分三种情形给出超奇异素数个数的无条件上界，并在实乘情形下进一步得到 Frobenius 迹及中间系数的分布上界。

论在 $\mathbb{Z}_p^2$-扩张上超奇异阿贝尔簇的 Mordell-Weil 秩

On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensions

本文研究在分裂素数 $p$ 处超奇异的阿贝尔簇沿虚二次域 $\mathbb{Z}_p^2$-扩张的 Mordell–Weil 秩增长，并讨论相关多符号 Selmer 群与 GSp(4) 的 $p$-进 $L$-函数。

On the parity of supersingular Weil polynomials

Let q be an odd power of a prime p and let A/Fq be a supersingular abelian variety of dimension g. We show that if p>2g+1, then the characteristic polynomial of the q-Frobenius is an even polynomial. This generalizes the well-known result on the vanishing of the trace of the p-Frobenius when p>3 for supersingular elliptic curves over Fp.

Finiteness of reductions of Hecke orbits

We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory an…

关于数域 $\mathbb{Z}_p$-扩张上 ${\rm GL}_2$-型 $p$-超奇异阿贝尔簇的 Tate-Shafarevich 群的增长

On the growth of Tate-Shafarevich groups of $p$-supersingular abelian varieties of ${\rm GL}_2$-type over $\mathbb{Z}_p$-extensions of number fields

研究具有实乘法的GL2型p-超奇异阿贝尔簇在数域Zp扩张上的Mordell-Weil秩有界性及Tate-Shafarevich群v-primary部分增长，构造局部点系与带符号Coleman映射，证明秩有界并给出渐近公式，推广了椭圆曲线情形。

具有坏约化的阿贝尔曲面与K3曲面的单值性结果

Monodromy results for abelian surfaces and K3 surfaces with bad reduction

本文证明了特征p下具有坏约化的常阿贝尔曲面与K3曲面的局部p进单值性定理，并由此在II型超奇异约化情形下得到Hecke轨道约化的有限性结果。

On fields of definition of components of the Siegel supersingular locus

Recently Ibukiyama proves an explicit formula for the number of certain non-principal polarizations on a superspecial abelian surface, extending his earlier work with Katsura for principal polarizations [Compos. Math. 1994]. As a consequence of Ibukiyama's formula, there exists a geometrically irreducible component of the Siegel supersingular locus which is defined over the prime finite field. In this note we give a direct proof of this result.

阿特金多项式用于具有实乘法的阿贝尔簇族

Atkin polynomials for families of abelian varieties with real multiplication

本文通过正交多项式与帕德逼近的关系，研究了具有实乘法的阿贝尔簇族的非普通域几何，并应用超几何恒等式确定了阿特金多项式的递推关系。

阿贝尔簇的同时超奇异约化

Simultaneous supersingular reductions of abelian varieties

本文研究了 Shimura 簇中阿贝尔概形的 $p$-adic Hecke 轨道在多个素数下的同时超奇异约化，精确刻画了映射像，并以此更概念化地证明了 Mazur 关于高阶 Heegner 点非挠性的猜想。

The supersingular loci and mass formulas on Siegel modular varieties

We describe the supersingular locus of the Siegel 3-fold with a parahoric level structure. We also study its higher dimensional generalization. Using this correspondence and a deep result of Li and Oort, we evaluate the number of irreducible components of the supersingular locus of the Siegel moduli space $\calA_{g,1,N}$ for arbitrary $g$.

Legendre Drinfeld modules and universal supersingular polynomials

We introduce a certain family of Drinfeld modules that we propose as analogues of the Legendre normal form elliptic curves. We exhibit explicit formulas for a certain period of such Drinfeld modules as well as formulas for the supersingular locus in that family, establishing a connection between these two kinds of formulas. Lastly, we also provide a closed formula for the supersingular polynomial in the $\jmath$-invariant for generic Drinfeld modules.

超奇异同构度分析中的精细洪堡不变量

Refined Humbert Invariants in Supersingular Isogeny Degree Analysis

研究主极化超特殊阿贝尔曲面的精细洪堡不变量，通过计算该不变量，在基于同构的密码学中实现检查极化同构、确定曲面几何类型、证明同构度上界及提供同构频率证据等应用，并从新视角分析固定同构度问题。

反圆塔扩张上p-超奇异椭圆曲线的Tate-Shafarevich群的渐近公式

Asymptotic formula for Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic extensions

本文在广义Heegner假设下，研究了p-超奇异椭圆曲线在反圆塔Z_p-扩张中数域上的Tate-Shafarevich群的p-准素子群，并给出了其渐近公式。

Dynamics on supersingular K3 surfaces

For any odd characteristic p=2 mod 3, we exhibit an explicit automorphism on the supersingular K3 surface of Artin invariant one which does not lift to any characteristic zero model. Our construction builds on elliptic fibrations to produce a closed formula for the automorphism's characteristic polynomial on second cohomology, which turns out to be an irreducible Salem polynomial of degree 22 with coefficients varying with p.

超奇异素数处CM椭圆曲线的Tate-Shafarevich群的渐近公式

An Asymptotic formula for Tate-Shafarevich groups of CM elliptic curves at supersingular primes

本文在Kobayashi符号Selmer群的附加假设下，对具有复乘的椭圆曲线在反圆$\mathbb{Z}_p$-扩张中间层上的Tate-Shafarevich群证明了渐近公式。

Supersingular curves on Picard modular surfaces modulo an inert prime

We study the supersingular curves on Picard modular surfaces modulo a prime $p$ which is inert in the underlying quadratic imaginary field. We analyze the automorphic vector bundles in characteristic $p$, and as an application derive a formula relating the number of irreducible components in the supersingular locus to the second Chern class of the surface.

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。

Exceptional splitting of reductions of abelian surfaces

Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces having real multiplication. Similar to Charles' theorem on exceptional isogeny of reductions of a given pair of elliptic curves and Elkies' theorem on supersingular reductions of a given elliptic curve, our theorem shows that a density-zero set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.

Elliptic Curves with Supersingular Reduction over $Γ$-extensions

This is a translation of a research announcement by Anas G. Nasybullin from 1976, in which he states formulas for the p-primary part of the Tate-Shafarevich group of an elliptic curve in cyclotomic $\Z_p$-extensions of number fields.

共轭主极化超特殊阿贝尔曲面之间的小次数同源

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。

关于阿贝尔簇的带符号Mordell-Weil群

On signed Mordell-Weil groups for abelian varieties

本文利用Coleman映射在超奇异阿贝尔簇上定义了多符号Mordell-Weil群，证明了一个弱控制定理，并引入了多符号Tate-Shafarevich群，推广了椭圆曲线的相关研究。

特征3上奇异椭圆曲线的分类

The Classification of Supersingular Elliptic Curves in Characteristic 3

本文在特征3上对奇异椭圆曲线进行显式分类，并提供点计数公式，以支持Hecke.jl中的实现。

实数域上具有四元数乘法的阿贝尔曲面的超特殊素数

Superspecial primes for QM abelian surfaces over real number fields

本文将Baba和Granath关于特定四元数乘法阿贝尔曲面超特殊素数的结果，扩展到模域为任意实嵌入数域且弱化局部条件的情形，证明依赖志村曲线Heegner除子相交理论。

关于I(1)-不变量：非阿贝尔海克代数情况

On the $I(1)$-invariants: Non-abelian Hecke algebra case

本文通过Iwahori-Hecke模型明确描述了universal module π_r的pro-p-Iwahori不变量，并确定了其作用，进而通过I(1)-不变量 functorially 恢复π_r并扩展了Ollivier的定理。

素数域上简单超特殊阿贝尔面的CM提升性

CM-liftability of simple superspecial abelian surfaces over prime fields

论文证明了在素数p>0的情况下，通过基变换成F_p^2，简单超特殊阿贝尔面可提升为CM型。

一维族中非一般曲线的质量公式

Mass formula for non-ordinary curves in one dimensional families

本文研究正特征下射影直线循环覆盖一维族中非一般曲线的质量公式，通过tautological intersection theory和$a$-numbers证明两个方程，推广了Eichler--Deuring等人的结果并在超椭圆曲线等新情形下确定了公式。

On counting certain abelian varieties over finite fields

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number $\sqrt{q}$. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of ge…

Tate 正规形式 $E_7$ 的 Hasse 不变量与 Fricke 群 $Γ_0^*(7)$ 的超奇异多项式

The Hasse invariant of the Tate normal form $E_7$ and the supersingular polynomial for the Fricke group $Γ_0^*(7)$

该研究用虚二次域类数刻画 Tate 正规形式 $E_7$ 的 Hasse 不变量在有限域上的因子分解，并推出 Fricke 群 $Γ_0^*(7)$ 超奇异多项式线性因子数的相关猜想。

On superspecial abelian surfaces and type numbers of totally definite quaternion algebras

In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field $\mathbb{F}_q$ of odd degree over $\mathbb{F}_p$ in the isogeny class corresponding to the Weil $q$-number $\pm\sqrt{q}$. This extends earlier works of T.-C. Yang and the present authors on the isomorphism classes of these abelian surfaces, and also generalizes the classical formula of Deuring for the number of endomorphism rings of supersingular elliptic curves. Our method is to explore the relationship between the type and class numbers of the quaternion orders concerned. We study the Pi…