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研究领域。一个典型的例子是著名的费马方程xn + yn = zn，其中n >。费马在1640年断言，这个方程在正整数中没有解，也就是说，两个n次幂的和永远不可能是n次幂，如果n + 0 + 2。1995年，怀尔斯利用弗雷和里贝特在算术几何中的思想和结果，证明了这个断言确实是正确的，这个问题得到了解决。***在研究这一领域的问题时，人们经常被引导到研究某些模空间的算术和几何：这些模空间是代数变体（如曲线、曲面等），其点对应于代数对象的同构类（如曲线）。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。怀尔斯证明中的一个关键技术是研究伽罗瓦表示（附在椭圆曲线上）并将它们与模形式联系起来。***本研究计划的目的是研究模空间ZN的算术和几何：这些曲面的点在某些椭圆曲线的伽罗瓦表示之间分类同构。这里特别感兴趣的是研究这些模曲面上的曲线，并识别那些来自模曲线的曲线。此外，检查位于两条这样的曲线交点上的点也是很有趣的。这种点是由椭圆曲线上的伽罗瓦表示的同构产生的，因此被称为CM点。***模空间ZN与一个称为Humbert曲面的模空间紧密相连，该曲面的点对具有n次椭圆子覆盖的2属曲线进行分类，因此，上述的一个主要应用是研究涉及Humbert曲面的问题。例如，研究这类亨伯特曲面的交点分量是一个可以在这里成功处理的问题。***这里的一种新技术可能被称为“逆算术几何”。这包括系统地使用数论中的方法和结果来推导关于某些模空间几何的有趣结果。***这项研究有许多应用，不仅在数论和算术几何，而且在代数几何（模空间，Humbert格式），数学物理（Hurwitz空间，模空间），动力系统（数学台球）和镜像对称。***此外，本研究计划涉及各级高素质人才（HQP）：暑期本科生（持有USRA），研究生（硕士和博士研究生）和博士后。