Schottky uniformization and Bloch-Wigner dilogarithm on higher genus curves

TL;DR

This paper proves the convergence of the Bloch-Wigner dilogarithm's defining iterated integral on higher genus Riemann surfaces, extending previous elliptic curve results using Schottky uniformization and Kleinian groups.

Contribution

It introduces a novel proof of convergence for the Bloch-Wigner function on higher genus curves, leveraging Schottky uniformization and recent advances in Kleinian groups.

Findings

Proves convergence of the Bloch-Wigner integral on higher genus curves.

Utilizes Schottky uniformization and recent results on Poincare metrics.

Extends classical elliptic curve results to higher genus cases.

Abstract

The question of convergence of iterated integrals on Riemann surfaces goes back to Bloch, Levin, and Zagier, who have proved this fact for various iterated integrals in the context of elliptic curves. In this work, I prove the convergence of the iterated integral defining Bloch-Wigner function on higher genus curves, utilizing some novel results of Hou in the theory of Schottky uniformization of Riemann surfaces, as well as some classical results of Bers and more recent results of Nayatani on Poincare metrics on domains of discontinuity of Kleinian groups.