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

这项研究将集中在算术的研究 三个单位群G上的自守形式的若干方面 关于CM扩展E/F的变量。 三个相关 主题将被调查。 第一个是调查 全纯函数的Fourier-Jacobi展开式中的系数 G上的自守形式，它是所有Hecke的特征函数 运营商 系数，这是θ函数上的一个 确定了E的复乘阿贝尔簇 由Hecke算子的特征值，但没有明确的 公式已知。 建议系数为 与L-函数的特殊值有关。 第二主题 关于G上的自守形式及其傅立叶变换的研究， 整数环上的Jacobi展开式。 第三个主题是 伽罗瓦表示相关的调查 G的自守表示 这项研究将集中在这些领域的数字 多元自守型理论与交换 品种 前者是编码大部分 要解决的数论问题的信息， 一个是在解决方案中使用强大的分析技术。 后者是与数字相关的几何对象 理论问题。 这些物体共同构成了强大的工具， 解决了数论中非常深奥的问题。 这个私家侦探是一 掌握这些工具和这么多的巨大利益将导致 从这项研究中。

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.