Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$

TL;DR

This paper explores the properties of multiple modular values, linking them to L-functions and automorphisms, advancing understanding of their algebraic and motivic structures within the context of modular forms.

Contribution

It establishes foundational properties of multiple modular values for the full modular group, connecting them to L-values and motivic Galois actions.

Findings

Relationship with special values of L-functions at positive integers

Action of the motivic Galois group via automorphisms

Properties established for the case of the full modular group

Abstract

Multiple modular values are a common generalisation of multiple zeta values and periods of modular forms, and are periods of a hypothetical Tannakian category of mixed modular motives. They are given by regularised iterated integrals on the upper half plane generalising the iterated Shimura integrals of Manin. In this paper, some first properties of the underlying theory are established in the case of the full modular group: in particular, the relationship with special values of L-functions of modular forms at all positive integers; and the action of the conjectural motivic Galois group via a certain group of automorphisms.