"Hurwitz spaces, Humbert schemes and modular curves"

本研究项目主要属于算术几何领域，即应用代数几何方法解决数论问题的领域。一个典型的例子是著名的Fermat方程x^n+y^n=z^n，其中n&gt；2。Fermat在1640年断言该方程没有正整数解，即，如果n&gt；2，则两个n次方和永远不可能是n次方。1995年，Wiles利用算术几何中Frey和Ribet的思想和结果证明了这一断言确实是正确的，这一问题得到了解决。 在研究这一领域的问题时，人们经常被引向对某些模空间的算术和几何的研究：这些是代数变体(如曲线、曲面等)。其点对代数对象(例如曲线)的同构类进行分类。例如，本提案标题中提到的Hurwitz空间、Humbert格式和模曲线都是各种类型的模空间的例子。 这项研究的目的是研究某些Hurwitz空间及其相关Humbert格式的算术和几何，特别是那些与模曲线(乘积)有关的算法和几何。这里特别感兴趣的是研究位于这种模空间中的曲线，并识别那些来自模曲线的曲线。 这里有一种新技术，可以称之为“逆算术几何”。这包括系统地使用数论中的方法和结果来得出关于某些模空间的几何的有趣结果。 这一研究不仅应用于数论和算术几何，而且还应用于代数几何(模空间、Humbert方案)、数学物理(Hurwitz空间、模空间)、动力系统(数学台球)和密码学(基于亏格2和3的曲线的密码系统)。

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).