Iterated integrals on the Legendre family of elliptic curves

TL;DR

This paper extends Chen's comparison isomorphism for iterated integrals to the Legendre family of elliptic curves, advancing algebraic-geometric methods for understanding periods.

Contribution

It develops an analogue of Chen's comparison isomorphism specifically for iterated integrals on the Legendre family of elliptic curves.

Findings

Establishes a new comparison isomorphism for elliptic curve periods

Bridges algebraic geometry and iterated integrals in elliptic settings

Provides tools for studying periods via iterated integrals on elliptic curves

Abstract

K.T. Chen showed that iterated integrals give comparison isomorphisms between the cohomologies of bar complexes and fundamental group rings. This led to the development of an algebraic-geometric approach to studying periods given by iterated integrals. In this paper we consider an analogue of this comparison isomorphism theorem for iterated integrals on the Legendre family.