Arithmetic of modular forms

主要研究人员和他的合作者建议研究朗兰兹程序和相关主题的某些方面。Serre的一个猜想预测了Q上Galois群的某些二维mod预示形式来自模形式，并进一步预测了形式的层次和重量。Serre的猜想可以被视为朗兰兹对应的mod p版本，权部分作为其在p处的局部行为的表现。所提出的研究的一个主要焦点涉及将Serre猜想的权重部分推广到Hilbert模形式和全实场的背景下，其中发展的公式揭示了新的现象。这项研究还涉及到与此相关的主题和应用，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。朗兰兹的程序预测了几何几何中的对象(多项式方程的解集，如椭圆曲线)和表示理论中的对象(具有对称属性的函数，如模形式)之间的深度对应。虽然最近取得了重大进展(例如，怀尔斯和莱夫格的进展)，但朗兰兹的计划在很大程度上仍然是猜测。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性质，这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.