Rational points, Galois representations, and fundamental groups

摘要奖Ellenberg DMS-0401616理性点，伽罗瓦表示，和基本群1。 调查员建议继续他的研究传统问题的算术几何（界限和渐近的数量合理点的品种，计数领域，计算莫德尔-韦伊秩的家庭交换品种）的手段不太传统的方法（伽罗瓦行动算术基本群体，非交换岩泽理论，Batyrev-Manin prostitics。2. 调查员的研究是在算术几何领域，其中心问题是确定一组解决方案ofequations。 这一领域的一个典型问题是费马猜想，它断言两个完全n次方不能和另一个n次方。 怀尔斯对这一猜想的解决源于对这一看似算术问题的深刻几何方法的应用;同样，研究者的研究集中在现代几何在方程及其整数解问题上的应用。

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.