Applications of Topology to Arithmetic and Algebraic Geometry

这个项目是将曲线的模空间的拓扑知识应用于算术和代数几何中的问题。该项目的三个主要焦点是：(1)了解普遍的椭圆动机。这些都是与所有椭圆曲线相关的动机(混合Hodge结构、Galois表示等)。例如，椭圆动机有助于解释为什么经典的模形式将关系强加于多个Zeta函数的特定值。它们还有助于PI理解与椭圆曲线相关的Galois表示的扩张与模形式之间的关系。(2)理解有限生成的无限域上光滑射影曲线的有理点。这是建立在首席调查员早期工作的基础上的。他和他的合作者正在将其推广到具有正特征的领域。(3)利用某些模空间的基本群(如映射类群)的知识来研究代数几何中的基本问题。PI正在研究的一个问题是，是否每条光滑的射影曲线都由光滑的平面曲线所支配。拓扑学是研究“拓扑空间”的形状。一组代数方程的解的集合构成了一个拓扑空间，它的形状往往对方程的解的集合起着很大的控制作用，特别是当解必须是整数或整数的比率(称为有理数)时。PI和他的合作者和学生正在利用他对一些非常特殊(和复杂)空间的形状的知识，称为模空间，来帮助理解一些非常一般的代数方程组的有理和整数解。长期以来，数学家一直在研究代数曲线及其有理点。他们的研究虽然深奥，但现在在现代密码学中扮演着重要的角色。这个项目由拓扑学计划和代数与数论计划联合资助。

This project is applying knowledge of the topology of moduli spaces of curves to problems in arithmetic and algebraic geometry. The three main foci of the project are to:(1) understand universal elliptic motives. These are motives (mixed Hodge structures, Galois representations, etc) associated to all elliptic curves. Elliptic motives are helping to explain, for example, why classical modular forms impose relations on special values of multiple zeta functions. They are also helping the PI to understand relations between extensions of Galois representations associated with elliptic curves and modular forms.(2) understand rational points of smooth projective curves over finitely generated infinite fields. This builds on earlier work of the principal investigator. He and his collaborators are extending it to fields of positive characteristic.(3) use knowledge of fundamental groups of certain moduli spaces (such as mapping class groups) to investigate fundamental questions in algebraic geometry. One such question being studied by the PI is whether every smooth projective curve is dominated by a smooth plane curve.Topology is the study of the shape of "topological spaces". The set of solutions of a set of algebraic equations forms a topological space.Its shape often exerts a lot of control over the set of solutions of the equations, especially when the solutions are required to be whole numbers, or ratios of whole numbers (known as "rational numbers"). The PI, with his collaborators and students, is using his knowledge of the shapes of some very special (and complicated) spaces, called moduli spaces, to help understand rational and integer solutions to some very general systems of algebraic equations. Algebraic curves and their rational points have long been studied by mathematicians. Their study, although esoteric, now plays an important role in modern cryptography.This project is jointly funded by the Topology Program and the Algebra and Number Theory Program.