Arithmetics on Jacobian Varieties

雅可比簇的算术

• 批准号: 06452004
• 负责人: YAMAMOTO Yoshihiko
• 金额: $ 3.78万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (B)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1995
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-06452004/
• 关键词: Algebraic curve; Jacobian variety; Abelian variety; Elliptic curve; Division points; Galois group; 超楕円曲線; 等分方程式; 有理点群; ガロワ拡大

1. Classification of Jacobian varieties of hyperelliptic curves defined over finite fields : We tried to (1) classify hyperelliptic cureves of genus g defined overfinite field of order q then (2) classify Jacobian varieties of curves classifyed by (1) in specific cases. In case q = 3,5 all hyperelliptic curves of genus 2 are classified and get isogeny classes by computing congruence zeta fanction of each curve. Then we determied the endmorphism rings of each Jacobians.2. Construction of abelian varieties of with a rational point of large order : (1) Orders of specific points on Jacobian varieties of hperelliptic curves can be given by the periods of continued fractional expansion of corrsponding functions on the curves. (2) We found a series of hyperelliptic curves of geneus 2 over the rationalfield, whose Jacobian varieties have a point of order 23.3. Computation of Galois groups of the n-division fields of abelian varieties : For the Jacobian varieties of hyperelliptic curves of gunus 2,2 and 3-division equations are determied by computational-procession.4. Construction of algebraic curves with many ration points : We got some series of algebraic curves over funcion fields with many rational points, by using the n-division equation of Jacobian varieties.

1. 有限域上定义的超椭圆曲线的雅可比变体的分类：我们尝试（1）对 q 阶超有限域定义的属 g 的超椭圆曲线进行分类，然后（2）在特定情况下对由（1）分类的曲线的雅可比变体进行分类。在 q = 3,5 的情况下，所有属 2 的超椭圆曲线都被分类，并通过计算每条曲线的同余 zeta 函数来获得同源类。然后确定了各雅可比行列式的同态环。 2．具有大阶有理点的阿贝尔簇的构造： (1) 准椭圆曲线雅可比簇上特定点的阶可以由曲线上相应函数的连续分数展开的周期给出。 (2) 我们在有理域上发现了一系列属 2 的超椭圆曲线，其雅可比变体的阶数为 23.3。阿贝尔簇n分域伽罗瓦群的计算：对于古努斯超椭圆曲线的雅可比簇，通过计算处理确定了2，2和3分方程。 4．多有理点代数曲线的构造：利用雅可比簇的n整除方程，我们得到了多有理点函数域上的一系列代数曲线。

项目成果

Hiroyuki Ogawa: "Curves of genus 2 with a raTional Torsion divisor of order 23" Proc. of the Japan Acad.70. 295-298 (1994)

Hiroyuki Okawa：“具有 23 阶有理扭转因子的 2 型曲线”Proc。

Tu Quoc Thang Le: "Representations of the cathegory of integrals by Kontsevich's interated integral" Comm.in Math.Physics. vol.168. 535-562 (1995)

Tu Quoc Thang Le：“通过 Kontsevich 的交互积分表示积分类别”Comm.in Math.Physics。

Yutaka Hiramine: "Regular partial conical flocks" Bull.Belgian Math.Soc.vol.2. 419-433 (1995)

Yutaka Hiramine：“规则的部分圆锥形羊群”Bull.Belgian Math.Soc.vol.2。

Mitsuyuki Ochiai: "Subgraphs of W-graphs and the 3-parallel version polynomial invariants of links" Proc.of the Japan Academy. 70, (Ser.A). 267-270 (1994)

Mitsuyuki Ochiai：“W 图的子图和链接的 3 并行版本多项式不变量”Proc.of the Japan Academy。

Tu Quoc Thang Le: "Kontsevich's integral for the Homfly plynomial and relations between values of multiple zeta functions" Topology and its applications. vol.62. 193-206 (1995)

Tu Quoc Thang Le：“Homfly 多项式的 Kontsevich 积分以及多个 zeta 函数值之间的关系”拓扑及其应用。