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群群是尖黎曼曲面模空间的基本群群，在拓扑学和数学物理中得到研究。利用课题组长给出的代数曲线算术肖特基-芒福德均匀化理论，构造了算术几何范畴的Teichmueller群群。 2．利用1中的结果，我们验证了Grothendieck关于Galois表示和共形场论给出的Monodromy表示的Teichmueller群群的动机的猜想，即表明这些对象是由基本对象生成的。 3．我们明确地描述了4次椭圆正态曲线的Chow形式，并证明了某些射影代数簇承认Chow形式具有特殊的性质。 4．研究了一些阿贝尔数域的单位群、类数和整数环。 5．通过引入沿能级趋于无穷递增的高斯核，讨论了不经重整化的微扰Chern-Simons积分的无限级渐进性。 6．确定了Hermite型评价码的最小距离，并构建了部分迹范数码的基础。 7．利用复曲线，我们得到了曲面映射类群的热尔韦对称表示。此外，我们还确定了3维车把的映射类群的虚拟上同调维数和欧拉数。

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。

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

Tatsuji Tanaka：“论 4 次椭圆正态曲线的 Chow 形式”Tsukuba J.Math.. 24. 109-125 (2000)

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)