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

作为几何拓扑学在算术几何领域的应用，М。MatSumoto和R.Hain得到了如下结果：数域的绝对Galois群在曲线C的幂等基本群上的作用达到最大值的充要条件是与C的雅可比中的代数上循环C-C^-对应的Galois上同调上循环。最近，关于亏格g&gt；1曲线的映射类群的加权完备化的Galois作用被证明是晶型的和混合型的，而对于g=1，它不是混合型的.Tamagawa构造了DVR上具有混合特征的光滑曲线族的非奇性的一个分解.这篇文章发表在《RIMS》杂志上。这一结果将真Grothendieck猜想简化为仿射猜想。Mochizuki发展了范畴算术几何和Frobenoid理论，为ABC猜想提供了一种途径。Tsuzuki引入了刚性上同调的泛上同调可下降和唯一Rham可下降的概念，得到了Kedlaya的有限二分性的推广。Kimura发展了纯动机的态射的有限维度的概念，并构造了许多有限维动机的非平凡例子。Sigeyuki Morita研究了映射类群和幂零基本群的导子代数，得到了关于Soule元素像的上同调结果和猜想。这篇论文已在Proc上发表。交响乐。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产生器，充分利用了最近CPU的并行性。一篇论文被MCQMC2006会刊接受。

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.