Modular Galois representations

模伽罗瓦表示

• 批准号: 0840649
• 负责人: Chandrashekhar Khare
• 金额: $ 31万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0840649&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 的工作涉及两种不同类型的对称性，一种是自然代数对称性，另一种是解析对称性。第一类是在有理数上定义的多项式方程根的对称性，第二类是称为模形式的复杂解析函数所具有的对称性。令人惊讶的事实是，这两种不同类型的对称性之间存在联系。自高斯时代以来，这一直隐含在数论工作中，正如他的二次互反定律所集中体现的那样。作为朗兰兹计划的一部分，它已被明确地纳入现代数论的统一主题。 PI 最近与 J-P 的合作。温滕贝格尔对塞尔猜想的结果使该计划取得了具体进展。多项式方程根和模形式这两种不同类型的对称性之间的关系是一种影响了广泛的数学甚至物理学（例如弦理论）的某些部分的现象。这是一个非常古老的主题，在当前的数学研究中仍然引起共鸣，并且有望在未来很长一段时间内如此。