Problems in Automorphic Forms, Arithmetic and Geometry

Ramakrishnan的DMS-0402044裁决摘要 这个建议的总主题是研究在各种情况下普遍存在的尖点形式。提出的四个具体问题涉及这些基本对象的算术、分析、几何和群论方面。具体来说，第一个问题是关于某些四维Galois表示是否是不可约的，而这类结果在大于3的维数上是未知的。第二个问题是关于全纯尖形的扭曲L-值的精确平均，以及由此产生的类数和一个丢番图问题。第三个问题表明，在特殊但普遍的情况下（如Delta函数），应该有Calabi-Yau变种，它们在几何上实现了相关的动机，因此L-函数，它编码了Frobeniustraces的统计特性。第四个问题是关于允许对合 * 的约化群G的 *-自对偶表示，并要求研究G上尖点形式的某种符号。本项目将研究连续和完美的自守函数与离散对象（如整数及其构建块，即素数）之间的某些神秘而深刻的关系，这些关系表现出令人难以忘怀的统计性质。自同构函数将自然界中发生的迷人对称性编码为磁盘上的波形。其中离散的音调具有诱人的算术含义。数学家和物理学家经常从离散的数字集合中建立生成函数，而迫切的问题是要知道这些函数是否允许隐藏的对称性，也就是说，它们是否描述了自守函数的音调。如果他们这样做，那么奇迹般的好处就会出现。开发它们是一件值得奋进，还有许多金矿尚未被发现。

Abstract for award DMS-0402044 of Ramakrishnan The general theme of this proposal is the study of the ubiquitous cuspforms in various contexts. The four specific problems proposed deal with thearithmetic, analytic, geometric and group theoreticaspects of these fundamental objects. To elaborate, the first problem asksif certain four dimensional Galois representations attached to cusp formsare irreducible, and such results are unknown in dimensions bigger than 3.The second problem deals with obtaining *exact* averages of twisted L-valuesof holomorphic cusp forms, and the ensuing implications for class numbersand a diophantine question. The third problem suggests that in special butpervasive instances (like for the Delta function), there should beCalabi-Yau varieties which geometrically realize the associated motives andhence the L-functions, which encode the statistical properties of Frobeniustraces. The fourth problem is concerned with the *-selfdual representationsof reductive groups G admitting involutions *, and asks for a study of acertain sign attached to cusp forms on G.This project will investigate certain mysterious and deep relationsBetween automorphic functions, which are continuous and pulchritudinous, anddiscrete objects like the integers and their building blocks, namely theprime numbers, which exhibit haunting statistical properties. Automorphicfunctions encode enchanting symmetries occurring in nature like thewaveforms on a disk. The discrete tones therein have alluring arithmeticalmeanings. Often mathematicians and physicists build generating functionsout of discrete collections of numbers, and the pressing problem is to know ifthese functions admit hidden symmetries, that is, if they are describingthe tones of automorphic functions. If they do, then miraculous benefits emerge.Exploiting them is a worthy endeavor, and there are many gold mines yet to be discovered.