Rational points, Galois representations, and fundamental groups

Princeton University, Princeton NJ

Investigators

Abstract

Abstract for award of Ellenberg DMS-0401616 Rational points, Galois representations, and fundamental groups 1. 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 of equations. 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 application of deep geometric methods to this seemingly arithmetic problem; the investigator's research, similarly, centers on the application of modern geometry to questions about equations and their solutions in integers.