Zeta-like Multizeta Values for higher genus curves

TL;DR

This paper explores new relations between multizeta values in higher genus function fields, revealing unique mechanisms and proposing conjectures about their algebraic properties, which differ from the genus zero case.

Contribution

It introduces the first known relations between multizeta values in higher genus, highlighting a universal family and novel mechanisms distinct from the rational function field case.

Findings

Identified relations between multizeta values in higher genus

Discovered a potentially universal family of relations

Proposed conjectures on algebraic properties of zeta-like values

Abstract

We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or conjecturally equivalently algebraic). These are the first known relations between multizetas, which are not with prime field coefficients. We seem to have one universal family. We also find that interestingly the mechanism with which the relations work is quite different from the rational function field case, raising interesting questions about the expected motivic interpretation in higher genus. We provide some data in support of the guesses.