Arithmetic of Cohomological Modular Forms

9701017 HIDA在对模L值、p-进Hecke代数、Galois表示及其Selmer群的算术进行了几年的基础研究后，数学家现在可以开始以更复杂的方式建立这一理论了。在A.Wiles和R.Taylor证明了Shimura-Taniyama猜想之后，包括证明了Mazur关于Hecke代数的猜想和模Galois表示的变形环，现在有了有效的工具来处理非交换Selmer群，特别是伴随模Galois表示的猜想。该奖项将支持与塞尔默集团相关的五个项目的研究。从岩泽理论的观点来看，数学家需要在Hecke代数的谱上构造p-进解析L函数来提供描述此类Selmer群的工具(方案II)。由于伴随的L值通常是非临界的，因此需要找出如何计算L值的代数部分的方法，这种方法足够可行，即使在非临界的情况下也能将其直接与Selmer群相连(方案I)。由于Taylor和Wiles的工作现在被K.Fujiwara推广到Hilbert模的情形，所以我们可以尝试攻击伴随Selmer群的两个变量的主要猜想(方案III)。研究这个问题的两个关键工具是：(I)通过Hecke代数和形变环分析基的变化；(Ii)J.Tilouine和E.Urban构造的辛群的p-进近普通Hecke代数理论。因此，研究Galois表示上的函数式运算如何通过变形环和Hecke代数来反映是很自然的(方案IV)，并将该理论推广到更一般的群，也许是酉群(方案V)。这个项目属于算术几何的一般领域--一个融合了数论和几何这两个最古老的数学领域的学科。事实证明，这种结合非常有成效--最近解决了几代人面临的问题。在其众多后果中，有一种是新的纠错码。这种代码对于现代计算机(硬盘)和光盘都是必不可少的。

9701017 Hida After several years of basic research on the Arithmetic of modular L-values,p-adic Hecke algebras, Galois representations and their Selmer groups, number theorists can now start building up the theory in a little more sophisticated way. After the proof of the Shimura-Taniyama conjecture by A. Wiles and R. Taylor, which include the proof of Mazur's conjecture on Hecke algebras and deformation rings of modular Galois representations, there are now effective tools to deal with non-abelian Selmer groups; in particular, those of adjoint modular Galois representations. This award will support research into five projects connected to Selmer groups. From the view point of Iwasawa's theory, number theorists need to construct p-adic analytic L-functions on the spectrum of the Hecke algebra to supply tools to describe such Selmer groups(Project II). Since adjoint L-values are often non-critical, one needs to find out how to compute the algebraic part of the L-values in an automorphic way feasible enough to connect them directly to Selmer groups even in non-critical case (Project I).Since the work of Taylor and Wiles is now generalized to the Hilbert modular case by K. Fujiwara, one can try to attack the two variable main conjecture for the adjoint Selmer groups (Project III). Two key tools in studying this problem are: (i) analysis of base change via Hecke algebras and deformation rings, and (ii) theory of p-adic nearly ordinary Hecke algebras for symplectic groups constructed by J. Tilouine and E. Urban. Thus it is natural to study how functorial operations on Galois representations are reflected by deformation rings and Hecke algebras (Project IV) and to generalize the theory to more general groups, perhaps unitary groups (Project V). This project falls into the general area of arithmetic geometry -a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that w ithstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.