Problems in Automorphic Forms, Arithmetic and Geometry

自守形式、算术和几何问题

• 批准号: 0402044
• 负责人: Dinakar Ramakrishnan
• 金额: $ 21万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0402044&HistoricalAwards=false
• 关键词: Problems; Automorphic; Forms; Arithmetic; Geometry

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.

Ramakrishnan DMS-0402044 奖摘要 该提案的总体主题是研究各种背景下普遍存在的尖头形状。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。详细来说，第一个问题询问附加到尖点形式的某些四维伽罗瓦表示是否是不可约的，并且这样的结果在大于 3 的维度中是未知的。第二个问题涉及获得全纯尖点形式的扭曲 L 值的*精确*平均值，并且随之而来的对班级人数的影响和丢番图问题。第三个问题表明，在特殊但普遍的情况下（如 Delta 函数），应该存在 Calabi-Yau 变体，它们在几何上实现了相关的动机，从而产生了 L 函数，它编码了 Frobenius 迹线的统计特性。第四个问题涉及承认对合的还原群 G 的*-自对偶表示，并要求研究 G 上的尖点形式所附加的某些符号。该项目将研究自守函数之间某些神秘而深刻的关系，这些函数是连续的且美丽的，以及离散对象，如整数及其构建块，即素数，它们表现出令人难以忘怀的统计特性。自同构函数编码自然界中发生的迷人对称性，就像磁盘上的波形一样。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从离散的数字集合中构建生成函数，紧迫的问题是知道这些函数是否承认隐藏对称性，即它们是否描述自守函数的音调。如果它们这样做了，就会出现奇迹般的好处。开发它们是一项值得努力的事情，而且还有许多金矿有待发现。