"Hurwitz spaces, Humbert schemes and modular curves"

“赫尔维茨空间、亨伯特方案和模曲线”

• 批准号: 105361-2012
• 负责人: Kani, Ernst
• 金额: $ 0.87万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569035
• 关键词: Hurwitz; spaces; Humbert; schemes; modular

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).

这项研究计划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费马方程x^n + y^n = z^n，其中n > 2。费马在1640年断言，这个方程在正整数中没有解，即，如果n > 2，两个n次方的和永远不会是n次方。这是解决在1995年时，怀尔斯，使用的想法和成果弗雷和里贝特在算术几何，证明了这一主张确实是正确的。 在研究这一领域的问题时，人们经常会研究某些模空间的算术和几何：这些是代数簇（如曲线、曲面等）。其点分类代数对象（例如曲线）的同构类。例如，在这个建议的标题中提到的Hurwitz空间，Humbert方案和模曲线都是各种类型的模空间的例子。 本研究计划的目的是研究某些Hurwitz空间及其相关Humbert方案的算术和几何，特别是那些与模曲线（的乘积）相关的方案。这里特别感兴趣的是研究位于这样的模空间中的曲线，并确定那些来自模曲线。 这里有一种新颖的技术可以称为“逆算术几何”。这包括系统地使用数论中的方法和结果，以得出有关某些模空间几何的有趣结果。 这项研究有许多应用，不仅数论和算术几何，但也代数几何（模空间，亨伯特计划），数学物理（赫维茨空间，模空间），动力系统（数学台球）和密码学（密码系统的基础上曲线的亏格2和3）。