Rational points, Galois representations, and fundamental groups

Ellenberg dms -0401616的有理点、伽罗瓦表示和基本群的授予摘要。研究者建议用不那么传统的方法（算术基本群上的伽罗瓦作用、非阿贝尔的Iwasawa理论、batyrevv - manin启发法）继续他对算术几何的传统问题（变种上有理点的个数的界和渐近性、计数数域、计算阿贝尔变种族的莫德尔-韦尔秩）的研究。研究者的研究是在等差几何领域，其中心问题是方程解集的确定。这方面的一个典型问题是费马猜想，它断言两个完全的n次幂不能等于另一个n次幂。怀尔斯对这个猜想的解决是由于对这个看似算术的问题应用了深奥的几何方法；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.