Rational points, Galois representations, and fundamental groups

有理点、伽罗瓦表示和基本群

• 批准号: 0401616
• 负责人: Jordan Ellenberg
• 金额: --
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2005-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401616&HistoricalAwards=false
• 关键词: Rational; points; Galois; representations; fundamental

Abstract for award of Ellenberg DMS-0401616Rational points, Galois representations, and fundamental groups1. The investigator proposes to continue his study of traditional problems of arithmetic geometry (bounds and asymptotics for the number of rational points on varieties, counting number fields, computing Mordell-Weil ranks of families of abelian varieties) by means of less traditional methods (Galois actions on arithmetic fundamental groups, non-abelian Iwasawa theory, Batyrev-Manin heuristics.)2. The investigator's research is in the field of arithmetic geometry,whose central problem is the determination of the set of solutions ofequations. A typical problem in this area is the conjecture of Fermat,which asserts that two perfect nth powers cannot sum to another nth power. The resolution of this conjecture by Wiles resulted from the applicationof deep geometric methods to this seemingly arithmetic problem; theinvestigator's research, similarly, centers on the application of modern geometry to questions about equations and their solutions in integers.

Ellenberg DMS-0401616 获奖摘要：有理点、伽罗瓦表示和基本群 1。 研究者建议通过不太传统的方法（伽罗瓦行动）继续研究算术几何的传统问题（簇上有理点数量的界限和渐近、计数数域、计算阿贝尔簇族的莫德尔-韦尔等级）算术基本群、非阿贝尔岩泽理论、巴特列夫-马宁启发法。）2． 研究者的研究属于算术几何领域，其中心问题是方程组解的确定。 这方面的一个典型问题是费马猜想，它断言两个完美的n次方不能和另一个n次方相加。 怀尔斯这一猜想的解决，源于对这个看似算术问题的深入几何方法的应用；类似地，研究者的研究集中在现代几何学在有关方程及其整数解的问题上的应用。