Mathematical Sciences: Arithmetic of Automorphic Forms on Unitary Groups in Three Variables

数学科学：三变量酉群自守形式的算术

• 批准号: 8703288
• 负责人: Jonathan Rogawski
• 金额: $ 4.22万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1990-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8703288&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Arithmetic; Automorphic; Forms

This research will concentrate on the study of arithmetic aspects of automorphic forms on a unitary group G in three variables with respect to a CM extension E/F. Three related topics will be investigated. The first is an investigation of the coefficients in the Fourier-Jacobi expansion of a holomorphic automorphic form on G which is an eigenfunction of all Hecke operators. The coefficients, which are theta functions on an abelian variety with complex multiplication by E, are determined by the eigenvalues of the Hecke operators, but no explicit formula is known. It is suggested that the coefficients are related to special values of L-functions. The second topic concerns the study of automorphic forms on G and their Fourier- Jacobi expansions over rings of integers. The third topic is an investigation of the Galois representations associated to automorphic representations of G. This research will concentrate on those areas of number theory concerning automorphic forms of many variables and abelian varieties. The former are functions which encode much of the information on the number theoretic problem to be solved allowing one to use the powerful techniques of analysis in the solution. The latter are geometric objects which are relevant to the number theory problem. Together these objects form powerful tools to solve the very deep problems of number theory. This P.I. is a master of these tools and so much of great interest will result from this research.

这项研究将集中在关于CM扩张E/F的具有三个变量的酉群G上的自同构形的算术方面的研究。第一部分研究了全纯自同构形在G上的傅里叶-雅可比展开的系数，它是所有Hecke算子的本征函数。系数是复乘于E的阿贝尔族上的theta函数，由Hecke算子的本征值决定，但没有明确的公式。结果表明，这些系数与L函数的特殊取值有关。第二个主题是关于G上的自同构形及其在整数环上的傅立叶-雅可比展开的研究。第三个主题是与G的自同构表示相关的伽罗瓦表示的研究。这项研究将集中在数论中关于多变量的自同构形和阿贝尔簇的那些领域。前者是对要解决的数论问题的大量信息进行编码的函数，允许人们在解中使用强大的分析技术。后者是与数论问题相关的几何对象。这些对象一起形成了强大的工具来解决数论中非常深刻的问题。这位私家侦探是这些工具的大师，这项研究将产生极大的兴趣。