Galois representations, Moduli Spaces and Applications

这项研究计划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，这个方程在正整数中没有解，即，如果n > 2，两个n次方的和永远不会是n次方。这是解决在1995年时，怀尔斯，使用的想法和成果弗雷和里贝特在算术几何，证明了这一主张确实是正确的。 在研究这一领域的问题时，人们经常会研究某些模空间的算术和几何：这些是代数簇（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。 证明怀尔斯定理的一个关键技术是研究所谓的伽罗瓦表示（附在椭圆曲线上），并将它们与模形式联系起来。 本研究计划的目的是研究模空间ZN的算术和几何：这些是其点分类椭圆曲线的某些伽罗瓦表示之间的同构的表面。这里特别感兴趣的是研究这些模曲面上的曲线，并确定那些来自模曲线。此外，它是感兴趣的，以检查点，其中位于两个这样的曲线的交点。这样的点产生于附加到椭圆曲线上的伽罗瓦表示的同构与复数乘法（CM），因此被称为CM点。 模空间ZN与称为Humbert曲面的某个模空间紧密相连，该曲面的点将亏格为2的曲线分类为具有N次椭圆子覆盖的曲线。因此，上述的主要应用是研究涉及Humbert曲面的问题。例如，研究的组成部分的交叉等亨伯特曲面是一个问题，可以成功地处理这里。 这里有一种新颖的技术，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，以获得有关某些模空间几何的有趣结果。 这项研究有许多应用，不仅数论和算术几何，但也代数几何（模空间，亨伯特计划），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，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（包括硕士和博士）。学生）和博士后学生。

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.