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.
