Degeneration of closed Riemann surfaces and limits of Jacobi varieties

封闭黎曼曲面的退化和雅可比簇的极限

• 批准号: 14540163
• 负责人: ABE Yukitaka
• 金额: $ 2.24万
• 依托单位: Toyama University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540163/
• 关键词: closed Riemann surface; jacobi variety; theta constant; Satake compactification; moduli space; degenerate abelian function; モジュライ空間; ヤコビ多用体; 代数的加法定理

Any closed Riemann surface of genus 2 is regarded as a double covering surface over the one-dimensional complex projective space with 6 branch points. We fix one of the branch points at the infinity. We found out a way to represent ratios of differences of these branch points by theta constants. We investigated behaviour of the ratios when Jacobi varieties of closed Riemann surfaces tend to limits. Then we recognized which curves are regarded as limits of degeneration.We also considered limits of Jacobi varieties. Compactification of the moduli space of abelian varieties was first given by Satake in 1956. It is very natural to require that limits of abelian varieties have function fields as limits of abelian function fields. These function fields are just what Weierstrass stated. We could determine them. From this point of view, the Satake compactification is the most natural one. Then we considered limits of Jacobi varieties in the Satake compactification in the above.

属2属的任何封闭的Riemann表面都被视为具有6个分支点的一维复合物射击空间上的双覆盖表面。我们修复了无穷大的分支点之一。我们发现了一种用theta常数来表示这些分支点差异的方法。当闭合黎曼表面的雅各比品种倾向于限制时，我们研究了比率的行为。然后，我们认识到哪些曲线被认为是变性的限制。我们还考虑了雅各比品种的限制。 1956年，Satake首先给出了Abelian品种的模量空间的压缩。很自然地要求Abelian品种的限制具有功能场作为Abelian功能场的限制。这些功能字段只是Weierstrass所说的。我们可以确定它们。从这个角度来看，Satake紧凑型是最自然的。然后，我们考虑了上述jacobi品种的限制。

项目成果

Completely magic, lupe and diamond properties of number squares

数字方块的完全魔法、卢佩和钻石属性

• 发表时间: 2003
• 期刊: Math.J. Toyama Univ. 26
• 作者: Kazuo Azukawa

Universal functions on Stein manifolds

斯坦因流形上的通用函数

• 发表时间: 2004
• 期刊: J.Math.Soc.Japan 56
• 作者: Y.Abe;P.Zappa
• 通讯作者: P.Zappa

Construction of automorphic forms for ample factors of quasi-abelian varieties

拟阿贝尔簇充因子自守形式的构造

• 发表时间: 2003
• 期刊: Kyushu J.Math. 57
• 作者: Y.Abe

Yukitaka Abe, Paolo Zappa: "Universal functions on Stein manifolds"J.Math.Soc.Japan. 56・1. 31-43 (2004)

Yukitaka Abe，Paolo Zappa：“斯坦因流形上的通用函数”J.Math.Soc.Japan 56・1（2004）。

Yukitaka Abe, Paolo Zappa: "Universal functions on Stein manifolds"J.Math.Soc.Japan. 55・4(掲載予定). (2003)

Yukitaka Abe，Paolo Zappa：“Stein 流形上的通用函数”J.Math.Soc.Japan 55・4（待出版）。