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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.

我们研究算术几何使用nonabel不变，基本组。主要研究者证明了：伽罗瓦作用在射影直线减三点的基本群上的导子李代数是由Soule元生成的。这解决了Deligne和Ihara的生成猜想。结果发表在Compositio Mathematicae上。使用类似的建设，负责调查证明了以下结果的行动算术基本组的模空间的基本组的曲线。绝对伽罗瓦群在曲线C的幂幺基本群上的作用象是极大的，当且仅当C的Jacobian中的代数圈C-C^-在伽罗瓦上同调中具有l-adic Abel Jacobi映射的非挠象。结果将出现在J. Inst. Math. Jussieu。玉川从曲线的几何基本群出发，研究了正特征曲线的重构问题。他证明，这是可能的，如果属是零，是可能的有限同构类在一般情况下。结果发表在J. Alg。Mochizuki推广了从其基本群重构概型的Anabel几何，并建立了范畴算术几何理论，有望对ABC猜想做出贡献。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。结果发表在Rend上。Sem等Mat.帕多瓦大学。木村用对称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这一点。S.森田提出了一个很好的进展费伯猜想的上同调环的模空间。结果发表在Topology上。