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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.

我们用一个非贝尔不变量（基本群）来研究等差几何。首席研究者证明了伽罗瓦作用在射影线减去三点的基本群上的推导李代数是由索勒元生成的。这就解决了Deligne和Ihara的生成猜想。研究结果发表在《数学合成》杂志上。利用类似的构造，Head研究者证明了模空间的算术基群对曲线基群的作用。当且仅当C的雅可比矩阵的代数循环C-C^-在伽罗瓦上同调中具有l-进Abel Jacobi映射的非扭转像时，绝对伽罗瓦群作用于曲线C的单幂基群的像是极大的。结果将出现在《数学》杂志上。Jussieu。Tamagawa从曲线的几何基本群出发，研究了曲线在正特征情况下的重构。他证明了如果属为零是可能的，并且在一般情况下对有限同构类是可能的。该研究结果发表在《J. Alg》杂志上。几何学。Mochizuki推广了由其固有基群重构一个格式的可列几何，并构造了一个分类算术几何理论，有望对ABC猜想有很好的贡献。Tsuzuki建立了刚性上同调的下降理论，证明了其维数的有限性和权谱序列的退化性。研究结果发表在《趋势》杂志上。扫描电镜。帕多瓦大学。Kimura用对称积和外积定义了纯动机有限维数的概念，并在曲线的情况下证明了这一概念。Morita对模空间上同环的Faber猜想作了很好的推进。研究结果发表在《拓扑学》杂志上。