Hilbert Modular Forms and Galois Representations

Serre的一个猜想预言了Q上伽罗瓦群的某些2维模预表示是由模形式产生的，并且它进一步预言了模形式的水平和权.主要研究者和他的合作者正在证明Serre猜想可能推广到全真实的域上的Hilbert模形式和伽罗瓦群. Serre猜想及其推广可以被看作是Langlands方案的一部分，Langlands方案从几何学的角度预测了物体之间的深层对应关系（多项式方程的解集，如椭圆曲线）和来自表示论的对象（具有非线性性质的函数，如模形式）。 虽然最近有了重大进展（例如，研究者和合作者在怀尔斯关于费马大定理的工作的基础上，证明了所有有理椭圆曲线都是由模形式产生的），但朗兰兹的程序在很大程度上仍然是抽象的。 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性质，这些性质反过来又可以应用于密码学。

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.