Galois representations, Moduli Spaces and Applications

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Fermat方程xn yn=zn，其中n&gt；2。Fermat在1640年断言这个方程没有正整数解，即，如果n&gt；2，两个n次方和永远不可能是n次方。这个问题在1995年得到解决，当时Wiles利用算术几何中Frey和Ribet的思想和结果证明了这一断言确实是正确的。 在研究这一领域的问题时，人们经常被引向对某些模空间的算术和几何的研究：这些是代数变体(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。 Wiles证明中的一项关键技术是研究已知的伽罗瓦表示(附在椭圆曲线上)，并将它们与模形式联系起来。 这个研究项目的目的是研究模空间ZN的算术和几何：这些曲面的点分类椭圆曲线的某些伽罗瓦表示之间的同构。这里特别感兴趣的是研究位于这些模曲面上的曲线，并识别那些来自模曲线的曲线。此外，有趣的是检查位于两条这样的曲线的交点上的点。这些点是由椭圆曲线上的伽罗瓦表示的同构产生的，因此称为复乘(CM)点。 模空间ZN与一个称为Humbert曲面的模空间密切相关，它的点用N次椭圆子覆盖对亏格为2的曲线进行分类，因此，上面的一个主要应用是研究与Humbert曲面有关的问题。例如，研究这种Humbert曲面的交点的分量就是一个可以在这里成功处理的问题。 这里有一种新技术，可以称之为“逆算术几何”。这包括系统地使用数论中的方法和结果来得出关于某些模空间的几何的有趣结果。 这一研究不仅应用于数论和算术几何，而且还应用于代数几何(模空间、Humbert格式)、数学物理(Hurwitz空间、模空间)、动力系统(数学台球)和镜像对称性。 此外，这项研究计划涉及所有级别的高素质人员(HQP)：暑期本科生(持有USRA)、研究生(均为理科硕士)。和博士后)和博士后。

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.