Problems in Automorphic Forms, Arithmetic and Geometry

本提案的总体主题是研究各种语境中普遍存在的cuspform。提出的四个具体问题涉及这些基本对象的算术、解析、几何和群论方面。为了详细说明，第一个问题是问附在尖形上的某些四维伽罗瓦表示是否不可约，并且在大于3的维数中这样的结果是未知的。第二个问题处理的是获得全纯顶点形式的扭曲l值的*精确*平均值，以及对类数和丢芬图问题的后续含义。第三个问题表明，在特殊但普遍的情况下（如Delta函数），应该有beCalabi-Yau变体，它们在几何上实现了相关的动机，从而实现了l函数，它编码了frobeniustrace的统计特性。第四个问题是关于有对合的约化群G的*-自对偶表示，并要求研究G上尖形的某些符号。这个项目将研究连续的、具有连续性的自同构函数与具有令人难以置信的统计性质的离散对象（如整数及其组成部分，即素数）之间的某些神秘而深刻的关系。自同态函数编码自然界中出现的迷人的对称性，就像圆盘上的波形一样。其中离散的音调具有诱人的算术意义。数学家和物理学家经常用离散的数字集合来构建生成函数，紧迫的问题是要知道这些函数是否承认隐藏的对称性，也就是说，它们是否在描述自同构函数的音调。如果他们这样做了，那么神奇的好处就会出现。开采它们是一项值得的努力，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.