"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).

该研究计划主要属于算术几何形状的区域，即应用代数几何方法来解决数量理论中的问题的区域。这里的一个典型例子是著名的费马特方程x^n + y^n = z^n，其中n>2。在1640年断言，该方程在积极的整数中没有解决方案，即，两个n-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-powers的总和永远不会是n-th-th-th-the Power，如果N> 2。断言确实是正确的。 在研究该领域的问题时，经常导致研究某些模量空间的算术和几何形状：这些是代数品种（例如曲线，表面等），其点分类了代数对象的同构类别（例如，曲线（例如曲线））。例如，该提案标题中提到的Hurwitz空间，Humbert方案和模块化曲线都是各种类型的模量空间的示例。 该研究计划的目的是研究某些Hurwitz空间及其相关的Humbert方案的算术和几何形状，尤其是与模块化曲线（产物）相关的计划。这里特别感兴趣的是研究在这种模量空间中的曲线，并确定来自模块化曲线的曲线。 这是一种新颖的技术，就是所谓的“逆算术几何形状”。这包括方法的系统用法和数字理论的结果，以获取有关某些模量空间几何形状的有趣结果。 这项研究有许多应用，不仅在数字理论和算术几何形状上，而且对代数几何形状（Moduli Space，Humbert计划），对数学物理学（Hurwitz Space，Moduli Space），动态系统（数学台球）和Cryptossography（CryptoSystography（CryptoSystography）（基于Cryptosystographys of Cryptographyss基于Genus of Genus 2和3）。