Modular Galois representations

模伽罗瓦表示

• 批准号: 0653821
• 负责人: Chandrashekhar Khare
• 金额: $ 37.5万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2008-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653821&HistoricalAwards=false
• 关键词: Modular; Galois; representations

The PI's work is related to two different kinds of symmetries, one algebraic in naure and the other that is analytic. The first kind is symmetries of the roots of polynomial equations defined over the rationals, and the second kind is that which complex analytic functions called modular forms have. The astonishing fact is that there is a connection between these two different kinds of symmetry.This has been implicit in number theoretic work since the time of Gauss, as epitomised in his law of quadratic reciprocity. It has been made explicitly into a unifying theme of modern number theory as part of the Langlands program. The PI's recent work with J-P. Wintenberger on Serre's conjecture results in concrete progress in this program.The relationship between these two different kinds of symmetries, of roots of polynomial equations and of modular forms, is a phenomenon that has affected a wide spectrum of mathematics and even some parts of physics like string theory. It is a theme that is very old, still resonant in current mathematical research, and promises to be so for a long time to come.

PI的工作与两种不同的对称性有关，一种是自然代数对称，另一种是解析对称。第一类是定义在有理数上的多项式方程的根的对称性，第二类是称为模形式的复解析函数所具有的对称性。令人惊讶的事实是，这两种不同的对称性之间存在着联系，这一点从高斯时代起就隐含在数论工作中，体现在他的二次互反定律中。作为朗兰兹纲领的一部分，它已经明确地成为现代数论的一个统一主题。PI最近与J-P. Wintenberger在塞尔猜想上的工作使这个计划取得了具体进展。多项式方程根和模形式这两种不同对称之间的关系是一种影响了广泛数学甚至物理学某些部分（如弦理论）的现象。这是一个非常古老的主题，仍然在当前的数学研究中引起共鸣，并承诺在未来很长一段时间内都是如此。