Arithmetic of Automorphic Forms on Reductive Groups

当研究者在追求三年前向NSF提出并资助的拨款提案中所描述的目标时，他提出了一些可行的想法，这些想法可能有助于验证自同构形式算术的某些基本猜想。我们现在开始攻击，以使这些部分有效的猜测成为基于坚实基础的方法论。项目A描述了一种建立在志村变种上定义的自同构形式的p进族理论的方法，特别是那些携带阿贝尔变种的正则纤维系统的自同构形式。在建立了这种p进自同构形式的控制理论之后，构造与每个自同构形式族相关联的p进l函数是项目b的目标。这些p进l函数和p进族应该在Iwasawa对CM域的理论理解、自同构泛函的验证（如基变换，项目C）和阿贝变体的模块化问题（项目D）中得到一些应用。有可能将这种理论推广到与志村变异无关的代数群。在方案E中，讨论了一般线性群的情况。如果成功，即使在这种非全纯情况下，自同构l函数的许多合理性结果也有望得到。数论问题，虽然通常以一种基本的方式表述，但可能令人惊讶地难以解决。费马最后定理（实际上是费马在17世纪提出的一个猜想）终于在1995年被怀尔斯解决了，经过了350年最强大的数学家的严重攻击。在其求解中，模形式和自同构形式的算法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将本研究中使用的一些工具的适用性扩展到更一般的环境中，包括更多的几何对象：那些对解集（所谓的“阿贝尔变体”）具有正则代数（群）结构的代数方程进行分类的空间。每个阿贝尔变体都以某种方式（推测）与满足许多对称性的非常特定的函数相关（因此称为“自同构形式”）。这种分类问题（所谓的“模块化问题”）在维数为1的最简单情况下的解决是上述Wiles证明的关键要素。该奖项描述的目标之一是将这种“模块化”结果推广到高维阿贝尔变体。

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.