Arithmetic of modular forms

首席研究员和他的合作者建议在朗兰兹的计划和相关主题方面进行工作。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。塞尔猜想可以看作是朗兰兹对应的模p版本，其权重部分是其在p处的局部行为的表现。本文提出的研究的一个主要焦点是将塞尔猜想的权重部分推广到希尔伯特模形式和全实场的背景下，在那里发展的公式揭示了新的现象。本研究还讨论了与此有联系和应用的主题，包括模形式之间的同余、伽罗瓦表示的变形、全局朗兰兹对应和l函数的特殊值。Langlands的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，怀尔斯和拉弗格的进展），但朗兰兹的计划在很大程度上仍然是推测性的。它在两个看似不同的数学分支之间提供的联系揭示了整数的属性，而这些属性反过来又可以应用于密码学。

The principal investigator and his collaborators propose to work on aspects of Langlands' program and related topics.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.Serre's conjecture can be viewed as a mod p version of Langlands'correspondence, and the weight part as a manifestation of itslocal behavior at p. A main focus of the proposed research concernsa generalization of the weight part of Serre's conjecture to the context of Hilbert modular forms and totally real fields,where the developing formulation reveals new phenomena. The research also addresses topics to which this has connectionsand applications, including congruences between modular forms, deformations of Galois representations, the global Langlands'correspondence and special values of L-functions.Langlands' program predicts a deep correspondence between objects fromalgebraic geometry (solution sets of polynomial equations, such as elliptic curves) and objects from representation theory (functions with symmetry properties, such as modular forms). While there has been significant recent progress (for example, the advances of Wiles and of Lafforgue), Langlands' program remains largely conjectural. The links it provides between two seemingly different branches of mathematics reveal properties of the integers, which can, in turn, have applications to cryptography.