Special Semester Program on Automorphic Forms, Shimura Varieties and L-functions; January 1-May 31, 2003, Fields Institute, Toronto, Canada

自守形式、志村簇和 L 函数特别学期课程；

• 批准号: 0211133
• 负责人: Freydoon Shahidi
• 金额: $ 5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0211133&HistoricalAwards=false
• 关键词: Special; Semester; Program; Automorphic; Forms

The theory of automorphic forms is a wide and deep subject touching many areas of mathematics, such as number theory, harmonic analysis and geometry. Langlands' program is an ambitious plan to develop the theory of automorphic forms in a systematic way. It opened a new frontier in Mathematics, and has given new insights and techniques in solving old problems. In fact, the solution of Fermat's last theorem by Andrew Wiles is one of the achievements of the Langlands' program.There have been many exciting new developments in the theory recently: the global Langlands correspondence for GL(n) over a functiona field by L. Lafforgue, using the ideas of Drinfeld; the local Langlands correspondence for GL(n) over a p-adic field by M. Harris and R. Taylor, using Shimura varieties, and by G. Henniart, using L-functions; Langlands' functoriality for symmetric cube of cusp form of GL(2) by H. Kim and F. Shahidi, and symmetric fourth of cusp form of GL(2) by H. Kim, using The Langlands-Shahidi method and the converse theorem of Cogdell-Piatetski-Shapiro. It is an ideal time to have a special program in automorphic forms to review these new developments. These have far-reaching applications in classical number theory. Especially, Langlands' functoriality of symmetric cube and symmetric fourth have direct applications to analytic number theory. In fact, one of the goals of the program is to bring experts in automorphic forms and analytic number theory to find applications of automorphic forms in analytic number theory, vice versa. One of the most important mathematical ideas of the second half of the 19th century is that an analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, one might want to count the number of solutions to a sequence of equations and have the answers as a sequence. Mathematicians call these kinds of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kinds of analytic functions often provide the answers to the discrete problems. An L-function is a type of generating function formed out of data associated with either a geometric object that is related to number theory or with a kind of 'analytic' function, called an automorphic form, which are determined by groups of linear transformations. These L-functions provide the right kinds of analytic functions. In the past thirty years, the study of these examples of L-functions has been systematized into a branch of number theory. In recent years many people have found new ways that L-functions encode discrete information, though there is still much to be discovered. The chief purpose of this program is to bring together experts in analytic number theory, automorphic forms, and geometry so that we may find more of the ways that arithmetic information is encoded by L-functions.

自动形式的理论是一个广泛而深刻的主题，触及数学的许多领域，例如数字理论，谐波分析和几何形状。兰兰兹（Langlands）的计划是一个雄心勃勃的计划，旨在以系统的方式发展自动形式的理论。 它开辟了一个新的数学领域，并为解决旧问题提供了新的见解和技术。实际上，安德鲁·威尔斯（Andrew Wiles）对费马特（Fermat）的最后定理的解决方案是兰兰兹（Langlands）计划的成就之一。最近，该理论中有许多令人兴奋的新发展：GL（n）的全球兰兰兹通讯（N）在l Fucationa领域的全球兰德兰人通信。 Lafforgue，使用德林菲尔德的想法； M. Harris和R. Taylor在P-ADIC领域的局部Langlands对应，使用Shimura品种，以及G. Henniart，使用L功能； H. Kim和F. Shahidi的Langlands的尖尖缘形式的对称立方体的功能性，以及使用Langlands-Shahidi方法和H. kim的cusp形式的对称性四分之一cogdell-piatetski-shapiro。现在是拥有自动形式的特殊计划来审查这些新事态发展的理想时机。这些在经典数理论中具有深远的应用。尤其是，兰兰兹对称立方体和对称第四的功能具有直接应用于分析数理论。实际上，该计划的目标之一是将自动形式和分析数理论的专家带入专家，以在分析数理论中找到自动形式的应用，反之亦然。 19世纪下半叶最重要的数学思想之一是分析公式通常编码离散信息。例如，人们可能想计算特定方程的解决方案的数量，但发现解决方案很难找到。另一方面，人们可能想计算一个方程序列的解决方案数量，并将答案作为序列。数学家称这类问题为“离散”。微积分中出现的功能，对微积分工作得很好，不是“离散的”，而是“分析”。令人惊讶的是，正确的分析功能通常为离散问题提供答案。 L功能是与与数字理论相关的几何对象或与一种“分析”功能相关的数据形成的一种生成函数的类型，称为自动形式，由线性转换组确定。 这些L功能提供了正确的分析功能。在过去的三十年中，对L功能的这些实例的研究已被系统化为数字理论的分支。近年来，许多人发现了L功能编码离散信息的新方法，尽管还有很多可发现的信息。该计划的主要目的是将分析数理论，自态形式和几何形状的专家汇集在一起​​，以便我们可以找到更多由L功能编码的算术信息的方式。