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Geometry of Moduli Spaces and Galois Actions

模空间的几何和伽罗瓦作用

基本信息

批准号：
13440005
负责人：
MATSUMOTO Makoto
金额：
$ 4.67万
依托单位：
HIROSHIMA UNIVERSITY (2002-2003)Kyoto University (2001)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

We investigated arithmetic geometry using a nonabelian invariant, the fundamental group. Head investigator proved the following : the derivation Lie algebra of Galois action on the fundamental group of projective line minus three points is generated by the Soule elements. This solves the generation conjecture by Deligne and Ihara. The result was published in Compositio Mathematicae. Using a similar construction, Head investigator proved the following result on the action of the arithmetic fundamental group of the moduli space on the fundamental group of curves. The image of the action of the absolute Galois group on the unipotent fundamental group of a curve C is maximal, if and only if the algebraic cycle C-C^-in the Jacobian of C has non-torsion image by l-adic Abel Jacobi map in the Galois cohomology. The result is to appear in J. Inst. Math. Jussieu. Tamagawa researched the reconstruction of a curve in the positive characteristic case from its geometric fundamental groups. He proved that it is possible if genus is zero, and is possible up to finite isomorphic classes in a general case. The result was published in J. Alg. Geom. Mochizuki is generalizing the anabelian geometry which reconstructs a scheme from its etale fundamental group, and is constructing a theory of categorical arithmetic geometry, which is expected to have a good contribution to ABC conjecture. Tsuzuki established the descent theory of the rigid cohomology and prove the finiteness of its dimension and degeneration of the weight spectral sequences. The result is published in Rend. Sem. Mat. Univ Padova. Kimura defined the notion of finite dimensionality of pure motives using symmetric and exterior products, and proved them in the case of curves. S. Morita made a good advance towards Faber's conjecture on the cohomology ring of the moduli space. The result was published in Topology.

我们使用非交换不变量（基本群）研究了算术几何。首席研究员证明了：射影线减三点的基本群上伽罗瓦作用的李代数的推导是由 Soule 元素产生的。这解决了德利涅和伊原的代猜想。结果发表在《Compositio Mathematicae》上。使用类似的构造，首席研究员证明了模空间的算术基本群对曲线基本群的作用的以下结果。当且仅当 C 的雅可比行列式中的代数环 C-C^- 在伽罗瓦上同调中通过 l-进阿贝尔雅可比映射具有非挠像时，绝对伽罗瓦群在曲线 C 的单能基本群上的作用的像最大。结果将出现在《J. Inst.》杂志上。数学。朱西厄.玉川研究了正特征情况下曲线从其几何基本群的重建。他证明了如果属为零则这是可能的，并且在一般情况下最多有有限的同构类。结果发表在《J. Alg》上。吉姆.望月新一正在推广阿贝尔几何，从其etale基本群重建一个格式，并正在构建分类算术几何理论，预计会对ABC猜想做出很好的贡献。 Tsuzuki建立了刚性上同调的下降理论，并证明了其维数有限性和权谱序列的退化性。结果发表在《Rend》上。 SEM。垫。帕多瓦大学。木村使用对称积和外积定义了纯动机的有限维概念，并在曲线的情况下证明了它们。 S. Morita 在模空间上同调环上的 Faber 猜想方面取得了很大进展。结果发表在《拓扑》杂志上。