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Beyond the framework of classical arithmetic geometry-Zeta, arithmetic topology, and categorical arithmetic geometry

超越经典算术几何的框架——Zeta、算术拓扑和分类算术几何

基本信息

批准号：
16204002
负责人：
MATSUMOTO Makoto
金额：
$ 15.56万
依托单位：
Hiroshima University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

As an application of geometric topology in the area of arithmetic geometry, М. Matsumoto and R. Hain Obtained the, following result : The action of the absolute Galois group of a number field on the unipotent fundamental group of a curve C attains the maximum size, if and only if the Galois cohomology cocycle corresponding to the algebraic cocycle C-C^- in the jacobian of C. This was published in J. of Institute of Mathematics Jussieu. More recently, the Galois action on the weighted completion of the mapping class group of the genus g>1 curves is proved to be crystalline and of mixed Tate type ; Whereas for g=1 it is not mixed Tate.Tamagawa constructed a resolution of nonsingularities of smooth curve family over DVR with mixed characteristic. This was published in Publications of RIMS. This results reduces the proper Grothendieck conjecture to the affine one. Mochizuki developped the theory of categorical arithmetic geometry and frobenioids, which give an approach to the ABC conjecture.Tsuzuki introduced the notion of the universal cohomology descendability and unimversal de Rham descendability for rigid cohomology, and obtaind an extention of Kedlaya's finite dimerisionality.Kimura developed the notion of finite dimensionality of a morphism of pure motives, and constructed many non-trivial examples of finite dimensional motives.Sigeyuki Morita studied the mapping class group and the derivation algebra of nilpotent fundamental groups, and obtained cohomological results and conjectures on the image of Soule elements. The paper was published in Proc. Sympos. Pure Mathematics.Matsumoto utilized the geometry of formal power series and Galois theory, to designe a new fast pseudorandom number generator taking full advantage of paralellism of recent CPUs. A paper is accepted in Proceedings of MCQMC2006.

作为几何拓扑学在算术几何中的应用，本文给出了一个新的几何拓扑学方法。松本和R.海恩得到了如下结果：数域的绝对伽罗瓦群在曲线C的幂幺基本群上的作用达到最大尺寸，当且仅当C的雅可比行列式中的代数上圈C-C^-对应的伽罗瓦上同调上圈。这是发表在J.数学研究所Jussieu。最近，证明了亏格g>1曲线的映射类群的加权完备化的Galois作用是结晶的且是混合Tate型的，而当g=1时，它不是混合Tate型的.Tamagawa构造了DVR上具有混合特征的光滑曲线族的非奇异性分解.这篇文章发表在RIMS出版物上。这一结果将原Grothendieck猜想简化为仿射猜想。Mochizuki发展了范畴算术几何和Frobenioids理论，解决了ABC猜想; Tsuzuki引入了刚性上同调的泛上同调可降性和unimplemented de Rham可降性的概念，并得到了Kedlaya有限二分性的推广; Kimura发展了纯动机态射的有限维数的概念，Sigeyuki Morita研究了幂零基本群的映射类群和导子代数，得到了关于Soule元象的上同调结果和图解。该论文发表在Proc. Sympos上。纯数学。松本利用形式幂级数几何和伽罗瓦理论，设计了一种新的快速伪随机数发生器，充分利用了当前CPU的并行性。一篇论文被MCQMC 2006会议录接受。