Applications of Topology to Arithmetic and Algebraic Geometry

Duke University, Durham NC

Investigators

Abstract

This project is applying knowledge of the topology of moduli spaces of curves to problems in arithmetic and algebraic geometry. The three main foci of the project are to: (1) understand universal elliptic motives. These are motives (mixed Hodge structures, Galois representations, etc) associated to all elliptic curves. Elliptic motives are helping to explain, for example, why classical modular forms impose relations on special values of multiple zeta functions. They are also helping the PI to understand relations between extensions of Galois representations associated with elliptic curves and modular forms. (2) understand rational points of smooth projective curves over finitely generated infinite fields. This builds on earlier work of the principal investigator. He and his collaborators are extending it to fields of positive characteristic. (3) use knowledge of fundamental groups of certain moduli spaces (such as mapping class groups) to investigate fundamental questions in algebraic geometry. One such question being studied by the PI is whether every smooth projective curve is dominated by a smooth plane curve. Topology is the study of the shape of "topological spaces". The set of solutions of a set of algebraic equations forms a topological space. Its shape often exerts a lot of control over the set of solutions of the equations, especially when the solutions are required to be whole numbers, or ratios of whole numbers (known as "rational numbers"). The PI, with his collaborators and students, is using his knowledge of the shapes of some very special (and complicated) spaces, called moduli spaces, to help understand rational and integer solutions to some very general systems of algebraic equations. Algebraic curves and their rational points have long been studied by mathematicians. Their study, although esoteric, now plays an important role in modern cryptography. This project is jointly funded by the Topology Program and the Algebra and Number Theory Program.