Rational points, Galois representations, and fundamental groups

Ellenberg DMS-0401616合理点数、伽罗瓦表示法和基本组的颁奖摘要1。作者建议继续研究传统的算术几何问题(有理点在簇上的个数的界和渐近性、计数数域、计算交换簇族的Mordell-Weil秩等)，用非传统的方法(算术基本群上的Galois作用、非交换的岩泽理论、Batyrev-Manin启发式)。2.研究者的研究是在算术几何领域，其中心问题是方程解集的确定。这一领域的一个典型问题是费马猜想，该猜想断言两个完全n次方不能等于另一个n次方。Wiles对这一猜想的解决源于对这一看似算术问题的深度几何方法的应用；调查者的研究同样集中于现代几何对方程及其整数解问题的应用。

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.