Geometry of Moduli Spaces and Galois Actions

我们使用一个非交换不变量--基本群来研究算术几何。首席研究者证明了：射影直线减三点的基本群上的Galois作用的导子李代数是由Soule元素生成的。这解决了Deligne和Ihara的世代猜想。这项研究结果发表在《数学合成》杂志上。使用类似的构造，首席研究者证明了以下关于模空间的算术基本群在曲线基本群上的作用的结果。绝对伽罗华群在曲线C的幂等基本群上的作用的像是极大的，当且仅当C的雅可比矩阵中的代数圈C-C^-具有由伽罗华上同调中的L-adadAbel-Jacobi映射所成的非挠象.其结果是出现在J.Inst上。数学课。朱西厄。玉川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问题。他证明了如果亏格为零是可能的，并且在一般情况下直到有限个同构类都是可能的。这项研究结果发表在《华尔街日报》上。天啊。Mochizuki推广了Anabelian几何，它从它的基本群重构了一个方案，并正在构建一种范畴算术几何理论，这有望对ABC猜想做出很好的贡献。Tsuzuki建立了刚性上同调的下降理论，并证明了其维度的有限性和权谱序列的退化。研究结果发表在《Rend》杂志上。扫描电子显微镜。垫子。帕多瓦大学。木村利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概念。S.Morita在模空间的上同调环上向Faber猜想迈进了一大步。研究结果发表在《拓扑学》杂志上。

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.