Arithmetic of Cohomological Modular Forms

在对模l值、p进Hecke代数、伽罗瓦表示及其Selmer群的算术进行了几年的基础研究之后，数论学家们现在可以开始以一种更复杂的方式建立理论了。在A. Wiles和R. Taylor证明了Shimura-Taniyama猜想（其中包括对Hecke代数和模伽罗瓦表示的变形环的Mazur猜想的证明）之后，现在有了处理非阿贝塞尔默群的有效工具；特别是伴随模伽罗瓦表示。该奖项将支持与塞尔默集团相关的五个项目的研究。从Iwasawa的理论来看，数论学家需要在Hecke代数的谱上构造p进解析l函数，以提供描述此类Selmer群的工具（项目II）。由于伴随l值通常是非临界的，人们需要找出如何以一种自同态的方式计算l值的代数部分，这种自同态的方式足够可行，即使在非临界情况下也能将它们直接连接到Selmer群（项目1）。由于Taylor和Wiles的工作现在由K. Fujiwara推广到Hilbert模情况，人们可以尝试攻击伴随Selmer群的两个变量主猜想（方案III）。研究这一问题的两个关键工具是：(i)通过Hecke代数和变形环分析基的变化，（ii） J. Tilouine和E. Urban构造的辛群的p进近普通Hecke代数理论。因此，研究伽罗瓦表示上的泛函操作如何通过变形环和Hecke代数反映（计划IV），并将理论推广到更一般的群，也许是酉群（计划V），是很自然的。这个项目属于算术几何的一般领域，这门学科融合了数学中两个最古老的领域：数论和几何。事实证明，这种结合非常富有成效——最近解决了几代人经受住的问题。其诸多后果之一是新的纠错码。这些代码对现代计算机（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.