Date of Award

Spring 2021

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Goncharov, Alexander

Abstract

The main motive of this thesis is to study the action of the motivic Galois group on the motivic fundamental group of an algebraic curve X punctured at a finite set of points. The algebraic, geometric, and analytic aspects of this action are examined in two cases: for X=P1 and for X an elliptic curve. To study this action, we rely on motivic correlators, canonical elements in the fundamental Lie coalgebra of the category of mixed motives over a number field. We trace three themes: (1) The Lie coalgebra structure on the motivic correlators. Using combinatorial arguments with an injection of Hodge theory, we find general families of relations (double shuffle relations) on these elements. (2) The Hodge realization of this structure. The canonical real periods of the motivic correlators are the Hodge correlator functions, functions of several X-valued variables that are computed as Feynman integrals. We find new functional equations on the Hodge correlator integrals. The proofs of results related to themes (1) and (2) are closely intertwined. (3) The geometry of modular manifolds. The phenomenon that the complex computing cohomology of some locally symmetric space can be mapped to the standard cochain complex of the motivic Lie coalgebra had been observed in several cases. Our work on Bianchi hyperbolic threefolds and the motivic fundamental groups of CM elliptic curves is a variation on this theme.

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