Galois representations, Moduli Spaces and Applications

伽罗瓦表示、模空间和应用

• 批准号: RGPIN-2018-04544
• 负责人: Kani, Ernst
• 金额: $ 1.17万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=655054
• 关键词: Galois; representations; Moduli; Spaces; Applications

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 xn + yn = zn, 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 correspond to isomorphism classes of algebraic objects (e.g. curves). For example, the points of modular curves correspond to isomorphism classes of elliptic curves with extra structure.*** A key technique in the proof of Wiles is to study what are known as Galois representations (attached to elliptic curves) and to relate them to modular forms.*** The aim of this research program is to study the arithmetic and the geometry of the moduli spaces ZN: these are surfaces whose points classify isomorphisms between certain Galois representations of elliptic curves. Of special interest here is to study the curves that lie on these moduli surfaces and to identify those that come from modular curves. In addition, it is of interest to examine the points which lie on the intersection of two such curves. Such points arise from isomorphisms of Galois representations attached to elliptic curves with complex multiplication (CM) and hence are called CM points.*** The moduli space ZN is closely connected with a certain moduli space called a Humbert surface whose points classify curves of genus 2 with an elliptic subcover of degree N. Thus, a main application of the above is to study problems involving Humbert surfaces. For example, the study of the components of the intersection of such Humbert surfaces is a problem that can be treated successfully here.*** 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 mirror symmetry.*** In addition, this research proposal involves highly qualified personnel (HQP) of all levels: summer undergraduate students (holding an USRA), graduate students (both M.Sc. and Ph.D. students) and post-doctoral students.

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费马方程 xn + yn = zn，其中 n > 2。费马在 1640 年断言该方程无正整数解，即两个 n 次方之和永远不可能是 n - 次方，如果 n > 2。这个问题在 1995 年得到解决，当时 Wiles 利用 Frey 和 Ribet 在算术几何中的思想和结果证明了这个断言确实正确。***在这一领域，人们经常被引导到对某些模空间的算术和几何的研究：这些是代数变体（例如曲线、曲面等），其点对应于代数对象（例如曲线）的同构类。例如，模曲线的点对应于具有额外结构的椭圆曲线的同构类。*** 怀尔斯证明中的一项关键技术是研究所谓的伽罗瓦表示（附加到椭圆曲线）并将它们与模形式。*** 本研究计划的目的是研究模空间 ZN 的算术和几何：这些曲面的点对椭圆曲线的某些伽罗瓦表示之间的同构进行分类。这里特别感兴趣的是研究位于这些模曲面上的曲线并识别来自模曲线的曲线。此外，检查位于两条这样的曲线的交点上的点也很有趣。这些点源自附加于具有复数乘法 (CM) 的椭圆曲线的伽罗瓦表示的同构，因此称为 CM 点。*** 模空间 ZN 与称为亨伯特曲面的特定模空间紧密相连，该曲面的点对亏格曲线进行分类2 具有 N 次椭圆子覆盖。因此，上述的主要应用是研究涉及亨伯特曲面的问题。例如，对此类亨伯特曲面相交分量的研究就是一个可以在这里成功处理的问题。*** 这里的一项新技术可能被称为“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。*** 这项研究有很多应用，不仅适用于数论和算术几何，还适用于代数几何（模空间、亨伯特方案）、数学物理（赫维茨空间、模空间）、动力系统（数学台球）和镜像对称。*** 此外，该研究提案涉及高素质人员各级HQP：暑期本科生（持有USRA）、研究生（硕士和博士生）和博士后。