The arithmetic of certain submotives of products of modular curves

本课题属于算术几何领域，即应用代数几何方法研究丢番图方程的领域。丢番图方程的一个典型例子是著名的Fermat方程：x^n+y^n=z^n，其中n&gt；2。Fermat在1640年断言该方程没有正整数解，即，如果n&gt；2，则两个n次方和永远不可能是n次方。直到最近(1995)Wiles基于Frey的想法和Ribet的工作才解决了这个问题。因此，Frey、Ribet和Wiles使用的一项重要技术是研究椭圆曲线及其相关的Galois表示。这个项目研究对这些表示之间的同构进行分类的表面。这里的一个关键问题是了解哪些特定类型的曲线可以位于这些曲面上，而对猜想答案的证明将构成费马方程广泛推广的第一步。正如这项建议所指出的，这项研究通过Hurwitz空间(计算椭圆曲线的覆盖)和椭圆曲线密码学(ECC)应用于动力系统(台球)和数学物理(镜像对称)领域。

This project belongs to the area of Arithmetic Geometry, i.e. to the area which applies the methods of Algebraic Geometry to study Diophantine equations. A typical example of a Diophantine equation is the famous Fermat equation: x^n+y^n=z^n 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 only recently (1995) by the work of Wiles, based on ideas of Frey and work of Ribet.     An important technique used by Frey, Ribet and Wiles is the study of elliptic curves and their associated Galois representations. This project studies surfaces which classify isomorphisms between such representations. A key problem here is to understand which curves of a certain type can lie on these surfaces, and a proof of the conjectured answer would constitute a first step towards a vast generalization of Fermat's equation.     As is indicated in this proposal, this research has applications to the areas of dynamical systems (billiards) and mathematical physics (mirror symmetry) via Hurwitz spaces (counting covers of elliptic curves) and also to elliptic curve cryptography (ECC).