Yuuki Tadokoro
We prove some value of the harmonic volume for the Klein quartic $C$ is nonzero modulo ${1/2}\{mathbb Z}$, using special values of the generalized hypergeometric function ${}_3F_2$. This result tells us the algebraic cycle $C-C^-$ is not algebraically equivalent to zero in the Jacobian variety $J(C)$.
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We give uniformizations of the Klein quartic curve and the Fermat septic curve as Shimura curves parametrizing Abelian $6$-folds with endomorphisms $\ZZ[\z_7]$.
Irrationality of the general smooth quartic $3$-fold using intermediate Jacobians
本文通过证明Klein四次三维簇的中间雅可比簇非曲线雅可比簇乘积，利用Clemens-Griffiths准则简洁地证明了其及一般光滑四次三维簇的非理性。
We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed ``explicitly'', we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic numbers in R whose minimal polynomials over Q can be given exactly and whose decimal approximations can be given to as many places as is necessary to distinguish them from their conjugat…
We give an algebro-geometric construction of some of the non-arithmetic ball quotients constructed by the author, Parker and Paupert. The new construction reveals a relationship between the corresponding orbifold fundamental groups and the automorphism group of the Klein quartic, and also with groups constructed by Barthel-Hirzebruch-Höfer and Couwenberg-Heckman-Looijenga.
We investigate the 2-dimensional jacobian conjecture via Klein's program.
Jacobian elliptic fibrations on the generalized Inose quartic of Picard rank sixteen
本文研究 Picard 格包含 $H \oplus E_7(-1) \oplus E_7(-1)$ 的复代数 K3 曲面族，证明其一般成员恰有四个互不等价的 Jacobian 椭圆纤维化，并构造了显式线束。
We describe the singularities and related properties of the arrangement of 21 reducible polars of Klein's quartic, containing Klein's well-known arrangement of $21$ lines.
The Torelli group of a compact non-orientable Klein surface is the subgroup of the modular group consisting of the mapping classes that act trivially on the first homology group of the surface. We prove that if a surface has genus at least $3$, then the Torelli group acts fixed points free on the Teichmüller space of the surface. That gives an embedding of the Torelli space of a Klein surface in the Torelli space of its complex double. We also construct real tori associated to Klein surfaces, which we call the Jacobian of the surface. We prove that this Jacobian is isomorphic to a component of…
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 …
For each $n \geq 3$ the authors provide an $n$-dimensional rigid compact complex manifold of Kodaira dimension $1$. First they construct a series of singular quotients of products of $(n-1)$ Fermat curves with the Klein quartic, which are rigid. Then using toric geometry a suitable resolution of singularities is constructed and the deformation theories of the singular model and of the resolutions are compared, showing the rigidity of the resolutions.
Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surface $Y$ is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms on $Y$ quotiented by the torsion-free part of the first integral homology of $Y$. Denote by $X$ the double cover of $Y$ given by orientation. The Jacobian of $Y$ is identified with the space of all degree zero holomorphic line bundles $L$ over $X$ with the property that $L$ is isomorphic to $σ^*\bar{L}$, where $σ$ is the involution of $X$.
Special types of quartic surfaces were much studied objects during the 1860s. Quartics were thus very much in the air when Sophus Lie and Felix Klein first met in Berlin in 1869. As this study shows, such surfaces played a major role in their subsequent work, much of which centered on linear and quadratic line complexes. This mutual interest led them to a number of new results on the quartic surfaces of Steiner, Plücker, and Kummer, as well as various types of ruled quartics studied earlier by Cremona. This paper, which draws on unpublished archival sources as well as published work from the p…
Mirror partner for a Klein quartic polynomial
本文研究了Klein四次多项式与Landau-Ginzburg奇点的镜像关系，证明其镜像奇点为Klein四元群的Z/2Z扩张。
We construct jacobians of plane quartics without complex multiplication, using Del Pezzo surfaces of degree 2.
Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is the Jacobian of a smooth plane curve, we show how to associate to f a classical plane invariant. As straightforward consequences of these constructions, when g=3 and k is a subfield of the complex field, we obtain (i) a new proof of a formula of Klein linking the modular form χ_{18} to the square of the discriminant of plane quartics ; (ii) a proof that one ca…
This article shows explicit relation between fractional expressions of Schottky-Klein type for hyperelliptic $σ$-functions and a product of differences of the algebraic coordinates on each stratum of natural stratification in a hyperelliptic Jacobian.
We use explicit methods to study the 4-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of 4-torsion points. We calculate the Galois action, and show that the image of the mod 4 Galois representation is isomorphic to the dihedral group of order 8. As applications, we calculate the Mordell-Weil group of the Jacobian variety of the Fermat quartic over each subfield of the 8-th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the 8-th cyclotomic…
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。
We explicitly construct the Kummer variety associated to the Jacobian of a hyperelliptic curve of genus 3 that is defined over a field of characteristic not equal to 2 and has a Weierstraß point defined over the same field. We also construct homogeneous quartic polynomials on the Kummer variety and show that they represent the duplication map.
We prove that the general quartic double solid with $k\leq 7$ nodes does not admit a Chow theoretic decomposition of the diagonal, or equivalently has a nontrivial universal ${\rm CH}_0$ group. The same holds if we replace in this statement "Chow theoretic" by "cohomological". In particular, it is not stably rational. We also prove that the general quartic double solid with seven nodes does not admit a universal codimension 2 cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibres of its Jacobian by a family of 1-cycles. This…
We give an explicit description of all 168 quartic curves over the field of two elements that are isomorphic to the Klein curve over an algebraic extension. Some of the curves have been known for their small class number, others for attaining the maximal number of rational points.
Abelian function fields on Jacobian varieties
本文介绍了产生于代数曲线Jacobian簇上的Kleinian $\wp$-函数域，展示了其作为Jacobian和Kummer簇代数模型的性质及加法定律，并以平面代数曲线为例详细解释。
Let M(2;0,3) be the moduli space of rank-2 stable vector bundles with Chern classes c_1=0, c_2=3 on the Fano threefold X, the double solid of index two. We prove that the vector bundles obtained by Serre's construction from smooth elliptic quintic curves on X form an open part of an irreducible component M' of M(2;0,3) and that the Abel-Jacobi map F:M'-->J(X) into the intermediate Jacobian J(X) defined by the second Chern class is generically finite of degree 84 onto a translate of the theta divisor. We also prove that the family of elliptic quintics on a general X is irreducible and of dimens…
In this paper the ubiquity of the Igusa quartic $B \subset \mathbb P^4$ shows up again, this time related to the Prym map $\mathfrak p : \mathcal R_6 \to \mathcal A_5$. We introduce the moduli space $\mathcal X$ of those quartic threefolds $X$ cutting twice a quadratic section of $B$. A general $X$ is $30$-nodal and the intermediate Jacobian $J(X)$ of its natural desingularization is a $5$-dimensional p.p. abelian variety. Let $\frak j: \mathcal X \to \mathcal A_5$ be the period map sending $X$ to $J(X)$, in the paper we study $\frak j$ and its relation to $\frak p$. As is well known the degre…
Decomposition of the diagonal, intermediate Jacobians, and universal codimension-2 cycles in positive characteristic
本文将复三维簇稳定有理性的两类对角上同调分解障碍推广到正特征有理链连通三维簇，证明相关代数代表元的典范自对偶性，并指出特征>2时带7结点的非常一般四次双立体消解不存在泛余维2闭链类。
Determinantal formulae for Jacobian theta functions that go back to Klein are elaborated, via an idea due to Matone and Volpato. Also, the natural square roots of theta constants on the moduli space of curves whose existence was shown by Tsuyumine are proved to have a spinorial structure.
Genus-One Fibrations and the Jacobian of Linear Slices in the Quintic Equal-Sum Problem
本文研究五次等和方程在特定约束下的解结构，证明了雅可比纤维的秩界及非奇异特性的条件。
The affine ring A of the affine Jacobian variety of a hyperelliptic curve of genus 3 is studied as a D-module. The conjecture on the minimal D-free resolution previously proposed is proved in this case. As a by-product a linear basis of A is explicitly constructed in terms of derivatives of Klein's hyperelliptic pe functions.
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