Generalized multiple zeta values over number fields I

TL;DR

This paper introduces a new framework for generalized multiple zeta values over number fields, inspired by Hodge theory and plectic principles, extending classical zeta values and polylogarithms to broader algebraic contexts.

Contribution

It constructs higher plectic Green functions and generalizes Hecke's formula for abelian L-functions over arbitrary number fields, offering a new approach to multiple zeta values.

Findings

Recovers classical multiple zeta values over 

Extends multiple polylogarithms at roots of unity to number fields

Provides a potential method for generalizing multiple zeta values

Abstract

Inspired by the theory of Hodge correlators due to Goncharov and by the plectic principle of Nekov\'a\v{r} and Scholl, we construct higher plectic Green functions and give a higher order generalization of Hecke's formula for abelian $L$-functions over arbitrary number fields. We hence provide a potential method to generalize multiple zeta values over number fields. We recover classical multiple zeta values and multiple polylogrithms evaluated at roots of unity, when the number field in consideration is the rational field $\mathbb{Q}$.