Arithmetic of Automorphic Forms on Reductive Groups

当这位研究人员追求三年前提出并资助给NSF的拨款提案中描述的目标时，他提出了一些可行的想法，这些想法可能有助于验证某些关于自同构形式算术的基本猜想。我们现在开始攻击，以使这些部分有效的推测成为一种建立在坚实基础上的方法。项目A描述了一种方法来建立定义在下村品种上的自同构型的p-进数族理论，特别是那些携带阿贝尔品种的规范纤维系统的品种。在建立了这种p-进自同构型的控制理论之后，构造与每一族自同构形相关的p-进L-函数是B计划的目标。这种p-进L函数和p-进数族在岩泽理论对CM领域的理解、自同构函数性的验证(如基变换；C方案)和交换变种的模性问题(方案D)中应该有一些应用。有可能将这种理论推广到与Shimura簇无关的代数群上。在项目E中，讨论了一般线性群的情况。如果成功，即使在这种非全纯的情况下，也有望得到许多自同构L函数的合理性结果。许多理论问题虽然通常是用初等的方式表述的，但可能会令人惊讶地难以解决。费马大定理(实际上是费马在17世纪提出的一个猜想)终于在1995年被威尔斯解决了，经受住了350年来最强大的数学家们的严重攻击。在其求解过程中，模形式和自同构形式的算法研究在许多方面起到了至关重要的作用。这里描述的程序旨在将这项工作中使用的一些工具的适用性扩大到更一般的环境，包括更多的几何对象：那些对其解集(所谓的阿贝尔簇)具有规范代数(群)结构的代数方程进行分类的空间。每一种阿贝尔变种都在某种程度上(猜想地)与满足多种对称性的一个非常特定的函数有关(因此它被称为“自同构形”)。在最简单的一维情形下，这种分类问题(所谓的“模性问题”)的解决是上述Wiles证明的一个关键部分。本奖项中描述的目标之一是将这种“模块化”结果推广到更高维度的阿贝尔品种。

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.