Arithmetic of Automorphic Forms on Reductive Groups

虽然研究人员正在追求的目标中所描述的赠款建议，以国家科学基金会提出和资助三年前，他已经提出了一些可行的想法，这可能是有用的，在验证一定数量的基本代数算术的自守形式。 我们现在开始攻击，以便使这些部分有效的推测成为基于坚实基础的方法论。 项目A描述了一种方法来建立理论的p-adic家庭的自守形式定义的志村品种，特别是那些携带一个典型的纤维系统的阿贝尔品种。 在建立了这种p-adic自守形式的控制理论之后，构造与每一族自守形式相关联的p-adic L-函数是项目B的目标。 应该有这样的p-adic L-函数和p-adic家庭的一些应用岩泽理论的理解CM领域，验证自守函（如基地变化;项目C）和模块化问题的交换品种（项目D）。 有一种可能性，这种理论延伸到代数群不相关的志村品种。 在项目E中，讨论了一般线性群的情况。 如果成功的话，许多自守L-函数的合理性结果甚至在这种非全纯的情况下也是可以预期的。数论问题，虽然经常以初等的方式表述，但可能非常难以解决。 费马最后定理（实际上是费马在世纪提出的一个猜想）在历经350年来数学家最强者的猛烈抨击后，终于在1995年被怀尔斯解决了。 在这一问题的解决过程中，模形式和自守形式的算术研究在许多方面起着至关重要的作用。 这里所描述的程序旨在扩大在这一奋进中使用的一些工具的适用性，以更一般的设置，包括更多的几何对象：那些空间分类代数方程的解集（所谓的“阿贝尔品种”）具有规范的代数（组）结构。 每一个阿贝尔簇都与一个满足许多对称性的非常特定的函数有某种联系（因此它被称为“自守形式”）。 这种分类问题（所谓的“模块化问题”）在最简单的一维情况下的解决方案是上述怀尔斯证明的关键组成部分。 在这个奖项中描述的目标之一是推广这种“模块化”的结果，以更高的维度阿贝尔品种。

While the investigator was pursuing the goals described in the grant proposal to NSF made and funded three years ago, he has come up with a few workable ideas which may be useful in verifying certain number of fundamental conjectures on arithmetic of automorphic forms. We now start an attack in order to make these partially valid speculations into a methodology based on a solid foundation. Project A describe a way to establish theory of p-adic families of automorphic forms defined on Shimura varieties, particularly those carrying a canonical fiber system of abelian varieties. After establishing such control theory of p-adic automorphic forms, construction of p-adic L-functions associated to each family of automorphic forms is the goal of Project B. There should be some applications of such p-adic L-functions and p-adic families to Iwasawa theoretic understanding of CM fields, to verification of automorphic functoriality (like base-change; Project C) and to modularity problems of abelian varieties (Project D). There is a possibility of extending such theory to algebraic groups not associated to Shimura varieties. In Project E, the case of general linear groups is discussed. If successful, many rationality results of automorphic L-functions are expected even in this non-holomorphic case.Number theoretic questions, though formulated often in an elementary way, could be astonishingly difficult to solve. The Fermat's last theorem (which was actually a conjecture made by Fermat in the seventeenth century) has been solved finally by Wiles in 1995, surviving for 350 years of serious attacks by the strongests of mathematicians. In its solution, arithmetic study of modular forms and automorphic forms played essential roles in many ways. The programs described here are intended to broaden the applicability of some of the tools used in this endeavor to more general setting, encompassing more geometric objects: those spaces classifying algebraic equations whose solution-set (so called "abelian varieties") having a canonical algebra (group) structure. Each abelian variety is somehow (conjecturally) related to a very specific function satisfying many symmetries (so it is called an "automorphic form"). The solution of such classification problems (so-called "modularity problem") in the simplest case of dimension one is a key ingredient of the above mentioned proof of Wiles. One of the goals described in this award is to generalize this "modularity" result to higher dimensional abelian varieties.