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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713091
• 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。