Motivic aspect of moduli space of algebraic curves

代数曲线模空间的动机方面

• 批准号: 11640035
• 负责人: ICHIKAWA Takashi
• 金额: $ 2.05万
• 依托单位: Saga University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640035/
• 关键词: Teichmueller groupoid; Moduli space; Riemann surface; Algebraic curve; Uniformization; Galois representation; Monodromy representation; Mapping class group; ガロア作用; チャーン・サイモンズ積分; 代数幾何符号

1. Teichmueller groupoids are fundamental groupoids of the moduli space of pointed Riemann surfaces, and are studied in topology and mathematical physics. Using the arithmetic Schottky-Mumford uniformization theory on algebraic curves given by the head investigator, we constructed Teichmueller groupoids in the category of arithmetic geometry.2. Using the result in 1, we verified Grothendieck's conjecture on motives attached to Teichmueller groupoids for Galois representations and monodromy representations given by conformal field theory, i.e. showed that these objects are generated by basic ones.3. We described explicitly the Chow-forms of elliptic normal curves of degree 4, and showed that certain projective algebraic varieties admit Chow-forms having a special property.4. We studied unit groups, class numbers and integer rings for some abelian number fields.5. We discussed the infinite level asymptotics of the perturbative Chern-Simons integral without renormalization, by introducing a Gaussian kernel increasing along the level tends to infinity.6. We determined the minimum distances of evaluation codes of the Hermite type, and constructed the bases of a part of trace-norm codes.7. Using complex of curves, we obtained Gervais' symmetric presentation for the mapping class group of a surface. Furthermore, we determined the virtual cohomological dimension and the Euler number of the mapping class group of a 3-dimensional handlebody.

1. Teichmueller群胚是点黎曼曲面的模空间的基本群胚，在拓扑学和数学物理学中有研究。利用首席研究员给出的代数曲线上的算术Schottky-Mumford单值化理论，构造了算术几何范畴内的Teichmueller群胚.利用文献[1]中的结果，我们证明了Grothendieck关于共形场论给出的Galois表示和monodromy表示的Teichmueller群胚的基元的猜想，即证明了这些对象是由基元生成的.我们明确地描述了四次椭圆法曲线的Chow-形式，并证明了某些射影代数簇允许具有特殊性质的Chow-形式.研究了某些交换数域的单位群、类数和整数环.通过引入一个沿着水平递增的高斯核，讨论了无重整化微扰Chern-Simons积分的无穷水平渐近性.确定了Hermite型求值码的最小距离，并构造了部分迹范数码的基.利用曲线的复形，得到了曲面映射类群的热尔韦对称表示。进一步，我们确定了一个3维非线性体的映射类群的虚上同调维数和Euler数。

项目成果

Itaru Mitoma: "Wiener space approach to a perturbative Chern-Simons integral"Stochastic Processes,Physics and Geometry,New Interplays,Proceedings. (to appear).

Itaru Mitoma：“微扰陈-西蒙斯积分的维纳空间方法”随机过程、物理和几何、新相互作用、论文集。

Tsuyoshi Uehara: "On minimum distance of algebraic-geometric codes"Adv.Stud.Contemp.Math.(Pusan). 1. 1-15 (1999)

Tsuyoshi Uehara：“论代数几何代码的最小距离”Adv.Stud.Contemp.Math.（釜山）。

Tatsuji Tanaka: "On the Chow-forms of elliptic normal curves of degree 4"Tsukuba J.Math.. vol.24. 109-125 (2000)

田中龙二：“论 4 次椭圆正态曲线的 Chow 形式”Tsukuba J.Math.. vol.24。

Toru Nakahara et al.: "On the units and the class number of certain composita of two quadratic fields"Proc.Japan Acad.. vol.75 A. 63-66 (1999)

Toru Nakahara 等：“关于两个二次域的某些组合的单位和类数”Proc.Japan Acad.. vol.75 A. 63-66 (1999)

Toru Nakahara et al.: "On the unit group and the class number of cetain composita of two real quadratic fields"Manuscripta Math.. (to appear).

Toru Nakahara 等人：“关于两个实二次域的某些组合的单位群和类数”Manuscripta Math..（待出版）。