Problems in Automorphic Forms, Arithmetic and Geometry

罗摩克里希南DMS-0402044奖的总主题是研究在各种背景下无处不在的尖形。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。具体地说，第一个问题是关于某些附加在尖点形式上的四维Galois表示是不可约的，并且这样的结果在大于3的维度上是未知的。第二个问题涉及得到全纯尖点形式扭曲的L-值的*精确*平均，以及随后对类数的影响和一个丢番图问题。第三个问题表明，在一些特殊但普遍的情况下(如Delta函数)，应该有Calabi-Yau变种几何地实现相关的动机，从而存在编码Frobenistraces统计性质的L函数。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上的尖形形式所附的某种符号。这个项目将研究连续和脉动的自同构函数与离散对象(如整数和它们的构件，即素数)之间的某些神秘而深刻的关系，它们表现出令人难忘的统计性质。自同构函数编码自然界中发生的迷人的对称，就像磁盘上的波形一样。其中的离散色调具有诱人的算术意义。数学家和物理学家经常从离散的数字集合中构造生成函数，迫切的问题是知道这些函数是否具有隐藏的对称性，即它们是否描述了自同构函数的音调。如果他们这样做了，那么奇迹般的好处就会出现。开采他们是一项值得努力的努力，还有许多金矿有待发现。

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.