Hilbert Modular Forms and Galois Representations

Serre的一个猜想预测了Q上Galois群的某些二维mod预示表示来自模形式，并进一步预测了这种形式的水平和重量.主要研究者和他的合作者正在研究Serre猜想在全实域上Hilbert模形式和Galois群的上下文中的一个可能的推广.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字证据。Serre的猜想及其推广可以被视为朗兰兹计划的一部分，该计划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(具有对称性质的函数，如模形式)之间的深度对应。虽然最近取得了重大进展(例如，研究人员和合作者在Wiles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猜测的。它提供了两个看似不同的数学分支之间的联系，揭示了整数的性质，反过来，整数可以应用于密码学。

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.