Hilbert Modular Forms and Galois Representations

希尔伯特模形式和伽罗瓦表示

• 批准号: 0303659
• 负责人: Fred Diamond
• 金额: $ 3.97万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-01-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0303659&HistoricalAwards=false
• 关键词: Hilbert; Modular; Forms; Galois; Representations

A conjecture of Serre predicts that certain 2-dimensional mod prepresentations of Galois groups over Q arise from modular forms,and it further predicts the level and weight of the form.The principal investigator and his collaborators are investigatinga possible generalization of Serre's conjecture to the contextof Hilbert modular forms and Galois groups over totally real fields. The prediction of the weights suggests interesting new phenomena, andthe immediate aims of the research are to investigate these phenomenaand provide numerical evidence for the generalization.Serre's conjecture and its generalizations can be viewed as part ofLanglands' program, which predicts a deep correspondence between objects fromalgebraic geometry (solution sets of polynomial equations, such as ellipticcurves) and objects from representation theory (functions with symmetryproperties, such as modular forms). While there has been significantrecent progress (for example, the investigator and collaborators,building on Wiles' work on Fermat's Last Theorem, proved that allrational elliptic curves arise from modular forms), Langlands'program remains largely conjectural. The links it provides betweentwo seemingly different branches of mathematics reveal properties of the integers, which can, in turn, have applications to cryptography.

Serre的一个猜想预测了Q上伽罗瓦群的某些二维模表示是由模形式产生的，并进一步预测了模形式的水平和权重。首席研究员和他的合作者正在研究将Serre猜想推广到希尔伯特模形式和伽罗瓦群在全实域上的可能性。权重的预测暗示了有趣的新现象，研究的直接目的是研究这些现象并为推广提供数值证据。Serre猜想及其推广可以看作是flanglands程序的一部分，该程序预测了代数几何对象（多项式方程的解集，如椭圆曲线）和表示理论对象（具有对称性质的函数，如模形式）之间的深度对应关系。虽然最近取得了重大进展（例如，研究人员和合作者在怀尔斯关于费马大定理的工作基础上，证明了所有有理椭圆曲线都是由模形式产生的），但朗兰兹的计划在很大程度上仍然是推测性的。它提供了两个看似不同的数学分支之间的联系，揭示了整数的属性，这些属性反过来又可以应用于密码学。