"Hurwitz spaces, Humbert schemes and modular curves"

点击翻译按钮获取中文摘要

This research program belongs mainly to the area of arithmetic geometry, i.e., to the area which applies the methods of algebraic geometry to solve problems in number theory. A typical example here is the famous Fermat equation x^n + y^n = z^n, where n > 2. Fermat asserted in 1640 that this equation has no solution in positive integers, i.e., that the sum of two n-th powers can never be an n-th power, if n > 2. This was resolved in 1995 when Wiles, using ideas and results of Frey and Ribet in arithmetic geometry, proved that this assertion is indeed true. In studying problems in this area, one is frequently led to the study of the arithmetic and geometry of certain moduli spaces: these are algebraic varieties (such as curves, surfaces, etc.) whose points classify isomorphism classes of algebraic objects (e.g. curves). For example, the Hurwitz spaces, Humbert schemes and modular curves mentioned in the title of this proposal are all examples of moduli spaces of various types. The aim of this research program is to study the arithmetic and geometry of certain Hurwitz spaces and of their associated Humbert schemes, particularly those that are related to (products of) modular curves. Of special interest here is to study the curves that lie in such moduli spaces and to identify those that come from modular curves. One novel technique here is what might be called "Inverse arithmetic geometry". This consists of the systematic usage of methods and results in number theory to derive interesting results about the geometry of certain moduli spaces. This research has many applications, not only to number theory and to arithmetic geometry, but also to algebraic geometry (moduli spaces, Humbert schemes), to mathematical physics(Hurwitz spaces, moduli spaces), to dynamical systems (mathematical billiards) and to cryptography (cryptosystems based on curves of genus 2 and 3).