Higher-genus multiple zeta values

TL;DR

This paper extends the concept of multiple zeta values to higher-genus Riemann surfaces, introducing a regularization method, exploring geometric relations, and identifying new algebraic relations among these values.

Contribution

It develops a canonical extension of multiple zeta values to higher-genus surfaces, providing a regularization scheme and uncovering new relations beyond known identities.

Findings

Regularization of higher-genus polylogarithms using Schottky uniformization.

Identification of relations among higher-genus multiple zeta values from geometric degeneration.

Discovery of new algebraic relations beyond classical polylogarithm identities.

Abstract

Multiple zeta values arise as special values of polylogarithms defined on Riemann surfaces of various genera. Building on the vast knowledge for classical and elliptic multiple zeta values, we explore a canonical extension of the formalism to Riemann surfaces of higher genera, which yields higher-genus multiple zeta values. We provide a regularization prescription for higher-genus polylogarithms, which we extend to higher-genus multiple zeta values. Our regularization uses the Schottky uniformization to trace back higher-genus endpoint regularization to known regularization at genus one. Additionally, we are commenting on relations among higher-genus multiple zeta values implied by degeneration of the underlying geometry, where we distinguish between the two types of separating and non-separating degeneration. Finally, employing functional relations for higher-genus polylogarithms in the Schottky uniformization, we explore relations among higher-genus multiple zeta values and check them against our numerical testing setup. We identify relations for higher-genus multiple zeta values beyond those implied by polylogarithm identities, thereby matching the situation for genus zero and genus one. While we find several known structures for elliptic multiple zeta values to generalize to relations for higher-genus multiple zeta values, there are further classes of relations arising from the interplay and combinatorics of different cycles.