Towards algebraic iterated integrals on elliptic curves via the universal vectorial extension

TL;DR

This paper explores algebraic iterated integrals on elliptic curves via the universal vectorial extension, connecting them to classical periods, multiple elliptic polylogarithms, and elliptic multiple zeta values, with implications for number theory.

Contribution

It introduces a framework for algebraic iterated integrals on elliptic curves using the universal vectorial extension, linking them to known special functions and periods.

Findings

Generalizes classical periods and quasi-periods of elliptic curves.

Connects iterated integrals to multiple elliptic polylogarithms.

Shows these integrals are periods in the sense of Kontsevich--Zagier.

Abstract

For an elliptic curve $E$ defined over a field $k\subset \mathbb C$, we study iterated path integrals of logarithmic differential forms on $E^\dagger$, the universal vectorial extension of $E$. These are generalizations of the classical periods and quasi-periods of $E$, and are closely related to multiple elliptic polylogarithms and elliptic multiple zeta values. Moreover, if $k$ is a finite extension of $\mathbb Q$, then these iterated integrals along paths between $k$-rational points are periods in the sense of Kontsevich--Zagier.