The fundamental group and extensions of motives of Jacobians of curves

TL;DR

This paper constructs new extensions of mixed Hodge structures related to the fundamental group of curves, linking motivic cycles in Jacobians to iterated integrals and generalizing previous results.

Contribution

It introduces a novel method to relate fundamental group structures to motivic cycle regulators in Jacobians, extending Colombo's theorem to broader classes of curves.

Findings

New iterated integral expression for regulators

Extensions of mixed Hodge structures from fundamental groups

Generalization of Colombo's theorem to non-hyperelliptic curves

Abstract

In this paper we construct extensions of mixed Hodge structures coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth, projective, pointed curve. These extensions correspond to the regulators of certain motivic cycles in the Jacobian of the curve which were constructed by Beilinson and Bloch. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of Colombo where she constructed the extension corresponding to Collino's cycles in the Jacobian of a hyperelliptic curve.