Galois Representations and Modular Forms

伽罗瓦表示和模形式

• 批准号: 0070659
• 负责人: Christopher Skinner
• 金额: $ 12.6万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070659&HistoricalAwards=false
• 关键词: Galois; Representations; Modular; Forms

0070659The proposer will investigate connections between Galois representations and modular forms in the context of two fundamental problems in algebraic number theory. The first of these problems is to determine which Galois representations are associated to automorphic representations. The second problem is to relate special values of L-functions of varieties or motives to associated algebraic quantities (such as orders of Selmer groups). In the direction of the first problem the proposer intends to pursue extensions of his recent work on the modularity of various families of two-dimensional p-adic Galois representations. In particular, the proposer intends to explore possible extensions to representations of higher dimension. In the direction of the second problem, the proposer will investigate the influence of divisibility properties of constant terms of Eisenstein series on higher-rank groups (which often involve special values of L-functions) on congruences between the Hecke eigenvalues of these Eisenstein series and those of cusp forms.Both of these problems fall under the rubric "Arithmetic Geometry," a branch of mathematics which attempts to apply sophisticated mathematics to the often easy-to-state, but-hard-to-solve problems of number theory. The problems most readily tackled by arithmetic geometry include 1) counting the number of integer solutions to systems of polynomials and 2) understanding L-functions (functions attached to systems of polynomials and which encode information about their solutions). These problems are of increasing importance in applications. Recent advances in computing, cryptography, and coding theory have depended on solutions to these problems as do many proposed advances in these areas.

0070659提议者将研究伽罗瓦表示和模形式之间的联系，在代数数论中的两个基本问题的背景下。 第一个问题是确定哪些伽罗瓦表示与自守表示有关。 第二个问题是将簇或基的L-函数的特殊值与相关的代数量（如塞尔默群的阶）联系起来。 在第一个问题的方向的提议者打算追求他最近的工作扩展的模块化的各种家庭的二维p-adic伽罗瓦表示。特别是，提议者打算探索更高维度的表示的可能扩展。 在第二个问题的方向上，提出者将考察爱森斯坦级数常数项的整除性质对高阶群的影响（这往往涉及特殊值的L-函数）之间的同余这些爱森斯坦级数的Hecke特征值和那些尖点形式。这两个问题属于标题“算术几何，数学的一个分支，试图将复杂的数学应用于通常容易陈述但难以解决的数论问题。 算术几何最容易解决的问题包括：1）计算多项式系统的整数解的数量; 2）理解L-函数（函数附加到多项式系统，并编码关于其解的信息）。 这些问题在应用中越来越重要。 计算、密码学和编码理论的最新进展依赖于这些问题的解决方案，这些领域的许多提议进展也是如此。