Quotients of Jacobians
Raisa Serova
本文证明亏格≥5的曲线雅可比簇在有限群作用下的商不是单直纹的，并分类了亏格3和4中商为单直纹的所有曲线。
我们证明：对于亏格大于或等于5的曲线，其雅可比簇在作用于该曲线的有限群作用下的商从不是单直纹的；并对亏格为3和4且其雅可比簇的商为单直纹的所有曲线进行了分类。
We prove that the quotient of Jacobian of a curve whose genus is greater than or equal to 5 under the action of a finite group acting on the curve is never uniruled, and classify all curves of genus 3 and 4 whose quotients of Jacobian is uniruled.
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Uniruledness of some low-dimensional ball quotients
本文利用复球上的反射模形式与 Gritsenko–Hulek 方法，证明若干3至5维球商单直纹，并给出 Hermitian 格及 P¹ 八点模空间等例子。
Quotients of Jacobians
本文研究了曲线的Jacobi簇在有限自同构群作用下的商，证明了亏格大于20时商具有典范奇点且Kodaira维数为0，并给出了亏格小于5时商为单有理的例子。
Rational curves on quotients of abelian varieties by finite groups
文[3]证明了交换阿贝尔簇$A$与有限阶自同构$g$的商是非圆的当且仅当$g$的某个幂满足一个数值条件$0<\age(g^k)<1$ .本文证明了$\age(g^k)=1$足以保证$A/\langle g\rangle$至少有一条有理曲线。
We show how to efficiently compute functions on jacobian varieties and their quotients. We deduce a quasi-optimal algorithm to compute $(l,l)$ isogenies between jacobians of genus two curves.
Non-torsion algebraic cycles on the Jacobians of Fermat quotients
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Jacobians with complex multiplication
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Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves
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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.
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Klein's simple group $H$ of order $168$ is the automorphism group of the plane quartic curve $C$, called Klein quartic. By Torelli Theorem, the full automorphism group $G$ of the Jacobian $J=J(C)$ is the group of order $336$, obtained by adding minus identity to $H$. The quotient variety $J/G$ can be alternatively represented as the quotient $\mathbb C^3/\tilde G$ of the complex $3$-space by the complex crystallographic group $\tilde G$, the extension of $G$ by the period lattice of the Klein quartic. Moreover, it turns out that $\tilde G$ is generated by affine complex reflections. According …
We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the Vélu's formula of genus 1. This work is based from the paper "Computing functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.
Fine compactified Jacobians
研究了节点曲线的Esteves紧致化雅可比算子。本文证明了对于节曲线$X$的单参数正则局部光滑，相对细紧化Jacobian的相对光滑轨迹同构于一般纤维Jacobian的Néron模型，我们证明了$X$的每个精细紧化雅可比行列式都允许根据部分的某些精细紧化雅可比行列式的分层此外，它可以实现为$X$的总爆破的合适的精细紧化雅可比矩阵的光滑轨迹的商。最后，我们确定当一个细紧雅可比同构于相应的Oda-Seshadri的粗紧雅可比。
The Intersection of Two Fermat Hypersurfaces in P^3 via Computation of Quotient Curves
研究了有限域上$\mathbb{P}^3$中两个特殊Fermat超曲面的交问题。利用Kani-Rosen分解，我们研究了这条曲线的算术性质。显式计算的导数是使用Gröbner基础算法。我们还研究了$p$ -秩，zeta函数，和有理点的数量，模$p$减少的曲线。我们证明亏格2商的雅可比矩阵是$(4,4)$ -分裂的.
Let $ϕ:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $ϕ$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois closure of the composition of $ϕ$ with a well-chosen map $Y\rightarrow \mathbb{P}^1$. This method allows us to recover all previously obtained descriptions of a Prym variety in terms of a Jacobian that are known to us, besides yielding new applications. We also find algebraic equations for some of these new cases, including one where $X$ has genus $3$, $Y$ has genu…
An Explicit Non-smoothable Component of the Compactified Jacobian
本文研究了非Gorenstein曲线的秩为1的模空间、无挠层或紧化Jacobian的分支。我们展示了一个通用的减少组件的尺寸等于算术亏格，并证明这是唯一的非光滑组件时，曲线有一个独特的奇异性，是有限的表示类型。证明了类似的结果的希尔伯特计划的点和报价计划参数化的对偶层。
Relations between tautological cycles on Jacobians
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Tjurina spectrum and graded symmetry of missing spectral numbers
研究超曲面孤立奇点的Tjurina子谱与缺失谱数的分级对称性，基于Jacobian环的自对偶性，改进了Briançon-Skoda指数估计。
On GIT quotients of Hilbert and Chow schemes of curves
本文的目的是公布关于P^{d-g}中d次亏格为g的曲线的Hilbert和Chow格式的GIT问题的一些结果，其全部细节将在随后的论文中出现。特别地，我们推广了L. Caporaso直到d>4（2g-2），我们观察到这是尖锐的。在2（2g-2）<d<7/2（2g-2）的范围内，我们得到了GIT商的全新描述。作为推论，我们得到了伪稳定曲线的模空间上泛雅可比矩阵的一个新的紧化。
Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions
本文证明了由反射生成的不可约复晶体反射群在三维复仿射空间上的商是权重为1,2,4,7的加权射影空间，其中相伴共线群是168阶克莱因单群，主要基于不变theta函数代数的计算。
Selmer varieties for curves with CM Jacobians
研究了定义在$\Q$上亏格至少为2的光滑射影曲线的$\Q_p$ -pro-unipotent基本群的标准商所对应的Selmer簇，其Jacobian分解为交换簇与复乘的乘积.利用初等多元Iwasawa理论证明了这类曲线的维数界，从而给出了$\Q$上丢番图有限性的一个新的证明.
Lefschetz properties and the Jacobian algebra of 3-dimensional hyperplane arrangements
本文研究高维商空间和1维几乎完全交的弱与强莱夫谢茨性质，并将结果应用于超平面构型的雅可比代数分析。
Simplicity of some Jacobians with many automorphisms
本文研究源于超椭圆曲线非分歧循环覆盖商曲线的雅可比簇族，通过形变论证证明其泛元是简单的，完全描述了其自同构代数与非极化自同构群，并据此证明了超椭圆曲线的 Prym 映射的泛单射性。
In this paper we construct extensions of mixed Hodge structures coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth, projective, pointed curve. These extensions correspond to the regulators of certain motivic cycles in the Jacobian of the curve which were constructed by Beilinson and Bloch. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of Colombo where she constructed the extension corresponding to Collino's cycles in the Jacobian of a hyperelliptic curve.
We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne-Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.
We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equation of the mirror $M$ of the orbifold ball quotient $(X,M)$ and by taking the quotient by an involution, we obtain an orbifold ball quotient surface with mirror birational to an interesting configuration of plane curves of degrees $1,2$ and $3$. We also exhibit an arrangement of f…
On rigid analytic uniformizations of Jacobians of Shimura curves
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Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. We compare three classifications of $BT_1$ group schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using words, canonical filtrations, and permutations. Using this comparison, we determine the Ekedahl--Oort types of Fermat quotient curves and we compute four invariants of the $p$-torsion group schemes of these curves.
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Torsion points on Jacobian varieties via Anderson's p-adic soliton theory
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Curves of genus 2 with group of automorphisms isomorphic to D_8 or D_12
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On the singularity of the Demjanenko matrix of quotients of Fermat curves
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Nontrivial elements of Sha explained through K3 surfaces
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Letter about component groups of elliptic curves
这是K写的一封信。里贝特呼叫J F. 1987年11月在梅斯特雷。这封信证明了三个公式，已aptured梅斯特和Oesterlé。这些公式涉及到作为某些模曲线的雅可比行列式的最优导数出现的椭圆曲线的分量组，以及分量组与雅可比行列式的复曲面分量的特征群之间的关系。(The“复曲面部件”和部件组以正特性计算。
Equations of Shimura curves of genus 2
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Chip-firing on graphs of groups
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Exact Quotients of Fresnel-Kummer Surfaces and Certified Biaxial Refraction
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We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We show that the only possibilities of genera of such curves are $2,3,4$ and $5$. We focus on the genus 5 case. We prove that up to translation, there is a unique hyperelliptic curve in the linear system of a general $(1,4)$ polarised abelian surface. Moreover, the curve is invariant with respect to a subgroup of translations isomorphic to the Klein group. We give the decomposition of the Jacobian of such a curve into abelian subvarieties displaying Jacobians of quotient curves and P…
Néron models of Jacobians over bases of arbitrary dimension
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K3 surfaces associated with curves of genus two
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Hyperelliptic Curves with Prescribed $p$-Torsion
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Genus 3 curves with many involutions and application to maximal curves in characteristic 2
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Finiteness results for modular curves of genus at least 2
有理数域Q上的曲线X是模曲线，如果对某个N，X被X_1（N）支配;另外，如果它在J_1（N）中的雅可比像包含在J_1（N）的新子簇中，则X称为新的模曲线。证明了对任意不小于2的整数g，亏格为g的Q上的新模曲线的集合是有限的且可计算的.对于可计算性的结果，我们证明了一个算法版本的de Franchis-Severi定理。对于有界性的新模曲线，对于雅可比矩阵是J_0（N）的新部分与N可被指定素数整除的商的新模曲线，以及对于水平在限制集中的模曲线（新的或非新的），证明了类似的有限性结果.我们详细研究了新的模超椭圆曲线。特别是，我们发现所有新的模曲线的亏格2明确，并构建可能是所有新的模超椭圆曲线的所有属的完整列表。最后证明了对特征为零的域k和至少为2的整数g，k上由Fermat曲线支配的亏格g曲线集是有限的且可计算的。
We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the way, we prove a criterion for the finiteness of rational points of degree $5$ on a curve of large genus over a number field using the results of Abramovich--Harris and Faltings on subvarieties of Jacobians.
We describe how the quadratic Chabauty method may be applied to explicitly determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell--Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or nontrivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin--Lehner quotients $X_0^+(N…
Noncongruence cusp forms, Veech groups and enumeration
本文证明某些 Veech 群存在权 2 模形式，其尖点傅里叶系数计数分支 Galois 覆盖，并构造非同余尖点形式计数亏格 2 覆盖，类比经典模曲线构造。
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