Modular forms and Galois representations in finite characteristic

有限特征中的模形式和伽罗瓦表示

• 批准号: 0355528
• 负责人: Chandrashekhar Khare
• 金额: --
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0355528&HistoricalAwards=false
• 关键词: Modular; forms; Galois; representations; finite

ABSTRACTSome of the most important theorems in mathematics are those whichconnect up different fields, concepts or view-points. In algebraicnumber theory many of such theorems go under the name of reciprocitylaws. The general framework for reciprocity laws is the Langlandsprogram. Much of the PI's work deals with issues that arise fromreciprocity laws especially in the context of linear, mod prepresentations of the absolute Galois group of the rationals. Areciprocity law in this case has been conjectured by J-P. Serre.Motivated by Serre's conjecture and related issues the PI has studiedcongruences between modular forms and deformations of Galoisrepresentations. The PI will continue to study and develop tools toapproach such reciprocity laws and related questions that have abearing on the intricate relationship between automorphic forms andGalois representations in the future.Any progress towards establishing reciprocity laws has a very broadimpact that is felt all across number theory. The work of the PI onreciprocity laws involves coming up with new ideas and techniques thatimpact many central areas of mathematics, like those mentioned above,and potentially might be useful to applications of number theory inareas like cryptography that are of great practical use. The PI alsoexpects to disseminate knowledge by involving students in researchproject

数学中一些最重要的定理是那些连接不同领域、概念或观点的定理。在代数数论中，许多这样的定理都被称为互反律。互惠律的一般框架是朗兰兹纲领。PI的大部分工作涉及的问题，所产生的fromreciprocity法律，特别是在线性的背景下，模prepresentations的绝对伽罗瓦集团的有理数。在这种情况下的一个互反律已由J-P. Serre证明，受Serre猜想及相关问题的启发，PI研究了Galois表示的模形式与变形之间的同余关系。未来，PI将继续研究和开发工具，以处理此类互易定律以及与自守形式和伽罗瓦表示之间错综复杂的关系有关的相关问题。建立互易定律的任何进展都将对整个数论产生非常广泛的影响。PI在互反定律上的工作涉及到提出新的思想和技术，这些思想和技术影响了数学的许多中心领域，就像上面提到的那些，并且可能对数论在密码学等领域的应用很有用。PI还希望通过让学生参与研究项目来传播知识