Universal Teichmuller Motives

代数曲线是由两个变量的多项式的零点定义的平面的子集。它们在几何学、物理学和数论中都很重要。曲线的模空间将给定拓扑类型的所有曲线参数化。这种类型由一个称为曲线亏格的整数来分类。通过回答亏格0和亏格1的问题，可以解决所有亏格曲线的模空间的一些问题。这一建议着重于理解曲线的模空间，特别是亏格1的模空间的拓扑与模空间的拓扑不变量的“算术对称”之间的相互作用。解决这些基本问题对于提高我们对整数和零点多项式集的拓扑对称性的理解是非常重要的。本课题的总体目标是了解所有亏格中曲线的模空间的基本群的完备化和路挠的动机方面。虽然关于亏格0曲线的模空间的路挠的基元结构已经被很好地理解(Deligne-Goncharov和Brown的工作)，但基本问题仍然存在，例如确定三次穿孔射影线的幂等基本群的自同构群中绝对Galois群的像的Zariski闭包(Grothendieck-Teichmuller程序的de~Rham版本)，以及理解为什么以及如何经典尖点形式在其深度过滤的相关等级中施加关系。PI的大部分注意力将集中在亏格1的情况上，因为它是最核心的，也因为它与经典模形式理论有联系。它通过退化到结点立方来影响亏格0的情形，并有助于解释为什么模形式对三次穿孔射影直线的幂等基本群上的伽罗瓦作用量施加条件。通过返回到Heller的拓扑学结果，可以将高亏格案例归结为亏格0和1。这个项目还将阐明Manin在迭代Shimura积分和椭圆KZB方程的算术方面的工作，这些工作出现在物理学中，但在这个项目中扮演着特殊的角色。

Algebraic curves are subsets of the plane defined by the vanishing of a polynomial of two variables. They are important in geometry, physics and number theory. Moduli spaces of curves parametrize all curves of a given topological type. This type is classified by a whole number called the genus of the curve. Some questions about moduli spaces of curves of all genera can be resolved by answering the the questions in genus zero and one. This proposal focuses on understanding the interaction between the topology of moduli spaces of curves, especially in genus one and the "arithmetic symmetries" of topological invariants of the moduli spaces. Resolving such basic questions is important in advancing our understanding of whole numbers and of the topological symmetries of zero sets of polynomials.The overall goal of this project is to understand motivic aspects of completions of fundamental groups and path torsors of moduli spaces of curves in all genera $\ge 0$. Although motivic structures on path torsors of moduli spaces of genus 0 curves are reasonably well understood (work of Deligne-Goncharov and Brown), fundamental problems remain, such as determining the Zariski closure of the image of the absolute Galois group in the automorphism group of the unipotent fundamental group of the thrice punctured projective line (a de~Rham version of the Grothendieck-Teichmuller program), and understanding why and how classical cusp forms impose relations in the associated graded of its depth filtration. Much of the PI's attention will be focused on the genus one case as it is the most central and also because of its connection to the theory of classical modular forms. It influences the genus 0 case by degeneration to the nodal cubic and should help explain why modular forms impose conditions on the Galois action on the unipotent fundamental group of the thrice punctured projective line. The higher genus cases can be reduced to the genus zero and one cases by results in topology that go back to Harer. This project will also clarify Manin's work on iterated Shimura integrals and arithmetic aspects of the elliptic KZB equation, which arose in physics, but plays a special role in this project.