p^k-torsion of genus two curves over F_{p^m}
Michael E. Zieve
确定了F_q上F_q-有理点群的阶可被q ^2整除的阿贝尔曲面的同构类。我们也解决了同样的问题的雅可比矩阵的亏格-2曲线。
We determine the isogeny classes of abelian surfaces over F_q whose group of F_q-rational points has order divisible by q^2. We also solve the same problem for Jacobians of genus-2 curves.
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p-torsion of Genus Two Curves Over Prime Fields of Characteristic p
考虑定义在特征为p的素域上具有复乘的超椭圆亏格2曲线的雅可比矩阵。本文证明了Jacobian的p-Sylow子群是平凡的或p阶的。
We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points ov…
Gluing Genus 1 and Genus 2 Curves Along $\ell$-torsion
本文提出一种系统搜索方法，寻找沿$\ell$-挠群粘合亏格1与亏格2曲线的素数及曲线候选对，改进了数值粘合算法，并发现了多个满足雅可比行列式同构条件的实例。
We establish a connection between torsion packets on curves of genus $2$ and pairs of elliptic curves realized as double covers of the projective line $\mathbb{P}_{x}^{1}$ that have many common torsion $x$-coordinates. This can be used to show that the set of common torsion $x$-coordinates has size at least $22$ infinitely often and has $34$ elements in some cases. We also explain how we obtained the current record example of a hyperelliptic torsion packet on a genus $2$ curve.
Let $X$ (resp. $Y$) be a curve of genus 1 (resp. 2) over a base field $k$ whose characteristic does not equal 2. We give criteria for the existence of a curve $Z$ over $k$ whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of $X$ and $Y$. Moreover, we give algorithms to construct the curve $Z$ once equations for $X$ and $Y$ are given. The first of these involves the use of hyperplane sections of the Kummer variety of $Y$ whose desingularization is isomorphic to $X$, whereas the second is based on interpolation methods involving numerical results over $\mathbb{C}$ …
A note on the ramification of torsion points lying on curves of genus at least two
设$C$是定义在具有代数闭剩余域的完全离散赋值环$R$的分式域$K$上的亏格$g\geqslant 2$的曲线.假设$\char(K)=0$和剩余域的特征不为2。假设Jacobian $\Jac(C)$在$R$上有半稳定约化.使用$K$ -有理点将$C$嵌入$\Jac(C)$中。我们证明了$C$上挠点的坐标位于由$\Jac(C)$上的$p$ -挠点的坐标在$K$上生成的场的唯一适度分歧的二次扩张中。
We define and compute the dressed elliptic genus of N = 2 heterotic compactifications with torsion that are principal two-torus bundles over a K3 surface. We consider the most general gauge bundle compatible with supersymmetry, a stable holomorphic vector bundle over the base together with an Abelian bundle over the total space, generalizing the computation previously done by the authors in the absence of the latter. Starting from a (0,2) gauged linear sigma-model with torsion we use supersymmetric localization to obtain the result. We provide also a mathematical definition of the dressed elli…
Rational torsion on simple genus two Jacobians
该研究针对有理数域上的二维亏格曲线，构造出阶数96等的新有理挠点子群，还记录了该类雅可比簇的所有已知有理挠点子群。
Genus-2 Jacobians with torsion points of large order
我们产生了新的显式例子亏格2曲线的有理数，其雅可比簇有合理的扭转点的大订单。特别是，我们产生了一个家庭的亏格2曲线在Q的雅可比矩阵有一个有理点的顺序48，参数化的秩2椭圆曲线在Q，我们展示了一个单一的亏格2曲线在Q的雅可比矩阵有一个有理点的顺序70，最大的顺序。我们还给出了新的例子，属2雅可比的有理点的顺序27，28和39。我们的大多数例子都是通过将两条椭圆曲线沿着它们的n-挠子群“粘合”在一起而产生的，其中n是2或3。2-胶合的例子来自作者15年前与Leprévost和Poonen共同开发的技术。3胶合的例子是可能的算法显式3胶合在非代数闭域最近开发的作者在联合工作与布鲁克，劳特，和Stevenstick。
Generators of Jacobians of Genus Two Curves
我们证明了在大多数与密码学相关的情况下，亏格为2的曲线的Jacobian上的Frobenius自同态可用关于l-扭点子群的适当基的对角矩阵表示.由此我们得到了l-挠点子群上Weil-配对的一个显式描述。最后，Weil配对的显式描述为我们提供了一个有效的，概率算法，以找到生成的子群的l-扭点的雅可比矩阵的亏格2曲线。
On the Frey-Mazur conjecture over low genus curves
Frey-Mazur猜想指出，$\mathbb{Q}$上的椭圆曲线由其对$p\geq 17$的$p$ -挠伽罗瓦表示确定到等距.我们研究了这个猜想的几何模拟，并表明，从Isobel类的“假椭圆曲线“--阿贝尔表面与四元数乘法--他们的$p$ -扭转伽罗瓦表示是一对一的功能领域的小亏格复杂曲线足够大的$p$相对于属。
Poincare sheaves on the moduli spaces of torsionfree sheaves over an irreducible curve
设$Y$是定义在实数上的几何不可约射影曲线。设$U_Y$（分别为$U'_Y$）是秩为$n$、度为$d$的$Y$上几何稳定的无挠层（分别为局部自由层）的模空间。定义$χ\, =\, d+n(1-\text{genus}(Y))$，其中$\text{genus}(Y)$是算术亏格。如果$2n$与$χ$互质，则存在$U_Y\times Y$上的庞加莱层。若$2n$与$χ$不互质，则在$U'_Y$的任何非空开子集上都不存在Poincare层。
We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.
On the Prym variety of genus 3 covers of genus 1 curves
本文针对亏格1曲线的非超椭圆亏格3覆盖，构造显式亏格2曲线，使雅可比簇同源于原曲线与覆盖曲线的雅可比簇乘积，推广了Nils Bruin的结果。
For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup.
Two-coverings of Jacobians of curves of genus two
我们找到了特征不同于2的任意基域上亏格为2的Jacobian的两个覆盖的显式方程。
Non-Cyclic Subgroups of Jacobians of Genus Two Curves
设E是定义在有限域上的椭圆曲线。Balasubramanian和Koblitz证明了：如果单位m_l的l次根不包含在基域中，则基域的一个域扩张包含m_l当且仅当E的l-扭点在同一个域扩张上有理。我们将这个结果推广到亏格为两条曲线的Jacobian。特别地，我们证明了在基域的相同域扩张上，Weil对和Tate对是非简并的.由此得到了超奇异亏格两条曲线的Jacobian的l-扭子群的完整刻画。特别地，我们证明了当l>3时，l-挠点在次数至多为24的域扩张上是有理的。
Optimal curves of genus 1,2 and 3
本文讨论了有限域上亏格为1，2，3的曲线的最大点数问题
For any number field, we prove that there exists a stacky curve of genus one half defined over the ring of its integers violating the local-global principle for integral points.
We prove that in the function field setting, $\ell$-torsion in the class groups of quadratic fields can be arbitrarily large. In fact, we explicitly produce a family whose $\ell$-rank growth matches the growth in the setting of genus theory, which might be best possible. We do this by specifically focusing on the Artin-Schreir curves $y^2=x^q-x$.
We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant.
We explain how we computed equations for all genus 4 curves defined of the field with 2 elements, up-to-isomorphism, and some of the data we obtained. We give descriptions also of nice models for genus 4 curves over characteristic 2 fields, in both the hyperelliptic case and the trigonal case.
Let $K$ be an algebraically closed field of characteristic different from $2$, $g$ a positive integer, $f(x)\in K[x]$ a degree $2g+1$ monic polynomial without repeated roots, $C_f: y^2=f(x)$ the corresponding genus g hyperelliptic curve over $K$, and $J$ the jacobian of $C_f$. We identify $C_f$ with the image of its canonical embedding into $J$ (the infinite point of $C_f$ goes to the zero of group law on $J$). It is known (arXiv:1809.03061 [math.AG]) that if $g>1$ then $C_f(K)$ does not contain torsion points, whose order lies between $3$ and $2g$. In this paper we study torsion points of ord…
We describe the ring of modular forms of degree 2 in characteristic 2 using its relation with curves of genus 2.
On Some Diophantine Parameters of the Cyclic Torsion Subgroups of Odd Order of Elliptic Curves over $\mathbb{Q}$
本文研究有理数域上椭圆曲线奇阶循环挠子群，并给出了其若干显式丢番图参数。
Genus formula for modular curves of $D$-elliptic sheaves
证明了$D$ -椭圆层的模曲线的亏格公式。我们利用这个公式证明了当$D$的判别式的阶趋于无穷大时，$D$ -椭圆层的模曲线的约化达到Drinfeld-Vladut界.
In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there are only finitely many torsion points whose coordinates are multiplicatively dependent. Moreover, we produce an effective result when the elliptic curve is defined over the rational numbers or has complex multiplication.
Curves of genus 3 over small finite fields
给出了基数为q的域上亏格为3的曲线上的最大有理点个数表，其中q<100。同时，对有限域上的Frobenius非经典四次型也作了一些注记.
The Number of Rational Points On Genus 4 Hyperelliptic Supersingular Curves in Characteristic 2
有限域上曲线领域的一个大问题是关于点的个数的分布：在亏格为$g$的曲线上，哪些数是点的个数？同样的问题也可以问到曲线的各种子类。本文对$2^n$，$n$奇阶有限域上亏格为4的超奇异超椭圆曲线上点数的可能性进行了分类.
Torsion of elliptic curves over cyclic cubic fields
本研究明确了循环三次域、非循环全实三次域与复三次域上椭圆曲线的全部可能挠群，填补了相关数论研究的空白。
Genus-2 curves and Jacobians with a given number of points
研究了有限域F上亏格为2的曲线C的有效构造问题，其中曲线C或其Jacobian具有给定数目N的F-有理点.在雅可比矩阵的情况下，我们表明，任何'CM建设'，以产生所需的属-2曲线必然需要时间指数的大小，其输入。另一方面，我们提供了一个算法，用于产生一个给定数量的点，多项式时间为大多数输入值的属2曲线。我们通过构造一条恰好有10^2014 + 9703（素数）点的亏格2曲线和两条恰好有10^2013点的亏格2曲线来说明该算法的实用性。在附录中，我们提供了一个完整的参数化，在一个任意的基域k的特征既不2也不3，家庭的亏格2曲线在k上，有k-有理度3映射到椭圆曲线，包括公式亏格2曲线，相关的椭圆曲线，和度3映射。
Torsion of rational elliptic curves over quadratic fields
设E是定义在Q上的椭圆曲线。研究了扭子群E（Q）_tors与扭子群E（K）_tors之间的关系，其中K是二次数域.
Torsion points on modular curves and Galois theory
给出了M的一个定理的证明. Baker和A. Tamagawa提出了一种新的方法，利用几乎有理扭点的概念，给出了嵌入其雅可比矩阵中的模曲线上的扭点的完整描述。'
The Euler characteristic of local systems on the moduli of curves and abelian varieties of genus three
我们展示了如何计算的欧拉示性数的局部系统相关联的不可约表示的辛群的亏格3上的模空间的曲线的亏格3和模空间的主要极化阿贝尔簇的3维。
The Geometric Bogomolov Conjecture for Small Genus Curves
博戈莫洛夫猜想是关于定义在整体域上的光滑完备曲线上的小高度代数点的有限性陈述。证明了特征为零的函数域上亏格至多为4的曲线的Bogomolov猜想的一个有效形式。我们恢复了已知的结果亏格2曲线，并在许多情况下，改进了已知的亏格3曲线的界限。对于亏格为4且约化坏约化的曲线，该猜想以前未被证明。
The Prym-Green Conjecture predicts that the resolution of a generic n-torsion paracanonical curve of every genus is natural. The conjecture has mostly been studied so far for level 2, that is, for Prym-canonical curves. Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove the Prym-Green Conjecture for curves of odd genus g and torsion bundles of sufficiently high order with respect to g. We also give partial results in even genus. In the process, we confirm the expectation of Barth and Verra concerning the number of curves in a fixe…
We exhibit a non-hyperelliptic curve C of genus 3 such that the class of the Ceresa cycle [C]-[-C] in the intermediate Jacobian of JC is torsion.
We exhibit a non-hyperelliptic curve C of genus 3 such that the class of the Ceresa cycle [C]-[(-1)*C] in JC modulo algebraic equivalence is torsion.
Teichmueller curves in genus three and just likely intersections in $G_m^n x G_a^n$
证明了紧致亏格为3的Riemann曲面的模空间只包含1/2条代数本原Teichmueller曲线.对于由亏格3中的全纯一形式组成的具有单个零点的地层，我们的有限性方法使用Teichmueller曲线上的Hodge丛的Harder-Narasimhan过滤来获得特征形零点位置的新信息。通过传递到模空间的边界，这给出了明确的限制Teichmueller曲线的尖点在一个射影线上的六个点的交比。这些约束类似于Zilber和Pink关于丢番图几何中不太可能相交的猜想中出现的约束。然而，在我们的例子中，一个是导致自然的交叉面与家庭的余维两个代数子群的$G_m^n \times G_a^n$（而不是更标准的$G_m^n$）。周围的代数群位于Zilber猜想的范围之外，但我们仍然能够证明一个足够强的高度界。对于亏格3中的一般层，我们通过计算机搜索$G_m^9$的余维2子簇的子环面，得到了整体挠序界。这些扭转界与新的边界的水平圆柱体的扭转阶的模量在这一层产生有限。中间层是用这些技术的混合处理的。
Discrete torsion, orbifold elliptic genera, and the chiral de Rham complex
给定一个紧致复代数簇，其有效作用为有限群$G$和类$α\in H^2(G,U(1))$，我们引入了一个具有离散挠率的轨道椭圆亏格$α$，记为$Ell^α_{orb}(X,G, q, y)$ .我们给这个属的解释，在连接到orbifold的$[X/G]$的手性德拉姆复杂。若$X$是Calabi-Yau型的，且$G$保持体积形式，则$Ell^α_{orb}(X,G, q, y)$是弱Jacobi形式.我们还得到了具有离散挠率的对称乘积的椭圆亏格的母函数的一个公式。
We observe that a lemma used in the study of even sets of nodes on surfaces applies almost verbatim to prove a celebrated formula of Gauss on the 2-torsion of the class group of a quadratic field.
Syzygies of torsion bundles and the geometry of the level l modular variety over M_g
我们制定，并在某些情况下证明，三个声明的纯度或更一般的自然性的决议，各种环可以附加到一个通用曲线的属g和扭点的顺序l在其雅可比矩阵。这些声明可以被视为一个类似的格林猜想，我们验证了它们的计算有界亏格。然后，我们计算相应的非零轨迹的上同调类的模空间R_{g,l}的扭曲的水平l曲线的亏格g和使用此导出结果的双有理几何的R_{g, l}。例如，当g> 11且R_{11,3}的Kodaira维数大于或等于19时，我们证明了R_{g,3}是一般型簇.在最后一节中，我们解释了概率的Prym-Green猜想在亏格8和水平2的意外失败。
A complete curve of genus 5 in the moduli space of curves of genus 3 and related questions
研究复数域上亏格3曲线模空间，通过特定方法证明其中存在一条亏格105的完备曲线。
Relating decision and search algorithms for rational points on curves of higher genus
对于定义在亏格至少为2的有理数上的仿射平面曲线，我们证明了解的存在性的判定算法也产生了所有解的搜索算法。
We give criteria for the Jacobian of a singular curve $X$ with at most ordinary $n$-point singularities to be anti-affine. In particular, for the case of curves with single ordinary double point we exhibit a relation with torsion divisors. If the geometric genus of the singular curve is atleast 3 and the normalization is non-hyperelliptic and non-bielliptic, then except for finitely many cases the Jacobian of $X$ is anti-affine. Furthermore, if the normalization is a general curve of genus atleast 3 then the Jacobian of $X$ is always anti-affine.
Higher rank Brill-Noether theory is completely known for curves of genus $\leq 3$. In this paper, we investigate the theory for curves of genus 4. Some of our results apply to curves of arbitrary genus.
The unirationality of the moduli space of curves of genus $\leq 14$
新的结果是亏格为14的曲线的模空间的单合理性。该方法适用于低级属。
The rationality of the moduli spaces of trigonal curves of odd genus
证明了奇亏格$g>4$三角曲线的模空间是有理的。
We study the moduli space of genus 3 hyperelliptic curves via the weighted projective space of binary octavics. This enables us to create a database of all genus 3 hyperelliptic curves defined over $\mathbb Q$, of weighted moduli height $\mathcal h =1$.
We study the induced Ekedahl-Oort stratification on the moduli of curves of genus 4 in positive characteristic.
We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.
A canonical curve of genus 17
我们计算方程的赫维茨曲线的亏格17，我们得出的结论是典型的理想的任何赫维茨曲线的亏格14或17是由二次曲面。
Families of genus two curves with many elliptic subcovers
我们确定所有的亏格2曲线，定义在$\mathbb C$，同时有2和3度椭圆子覆盖。这类曲线的轨迹在$\mathcal M_2$中有三个不可约的1维亏格零分量。对于每个分支，我们找到一个合理的参数化，并构造相应的亏格2曲线及其椭圆覆盖方程的参数化。这类亏格为2的曲线族是首次确定的。进一步证明了在$\mathbb Q$上只存在1/2个亏格为2的曲线（直到$\mathbb C$ -同构），它们具有同样定义在$\mathbb Q$上的2次和3次椭圆子覆盖.
We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree $p$, resp. $d$, onto smooth elliptic curves, with particular attention to the case $p$ prime.
On the rationality of the moduli of higher spin curves in low genus
高自旋曲线的模空间的整体几何和它们的双有理分类对于g >= 2和r > 2是未知的。利用相当相关的尺规作图法，我们几乎完成了亏格g <= 4的已知结果的图象，表明亏格3的偶、奇4-旋曲线、亏格4的奇旋曲线和亏格4的3-旋曲线的模空间的合理性。
Admissible constants for genus 2 curves
S.- W. Zhang最近在数域和函数域上引入了亏格至少为2的曲线的一个新的adelic不变量。当亏格等于2时，我们计算这个不变量。
Degree even coverings of elliptic curves by genus 2 curves
本文研究了偶数$n$的具有$(n, n)$ -分裂Jacobian的亏格2曲线。
Uniformizing Tropical Curves I: Genus Zero and One
在热带几何中，给定一个复曲面簇中的曲线，我们定义一个嵌入在欧几里得空间中的对应图。我们研究的问题，扭转这个过程的曲线亏格0和1。我们的方法侧重于描述曲线的参数化，而不是由他们的定义方程，我们给参数化的有理函数在属零的情况下，非阿基米德椭圆函数在属一的情况下。对于亏格为零的曲线，那些可以提升的图可以用完全组合的方式来刻画。对于亏格为1的曲线，证明了Mikhalkin给出的某些条件是充分的，并给出了一个新的必要条件。
Infinitesimal Torelli theorem for surfaces with c_1^2=3, χ=2, and the torsion group Z/3
本文证明了第一陈数为3，几何亏格为1，不规则性为0且具有非平凡3-扭除子的一般极小复曲面X的无穷小Torelli定理.我们还证明了具有上述不变量的曲面的粗模空间是一个14维单有理簇。
Genus 2 curve configurations on Fano surfaces
研究了三次三重Fano曲面上亏格为2的曲线的构型。我们建立了作用在这样的曲面S上的对合自同构与S上亏格为2的曲线之间的联系。我们根据这些对合生成的自同构群对Fano曲面进行了部分分类，并确定了它们亏格为2的曲线的构型。研究了Klein三次曲面的Fano曲面，其中55条亏格为2的曲线生成Néron-Severi群的秩为25=h^{1，1}的指数为2的子群.
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