Universal Families of Eulerian Multiple Zeta Values in Positive Characteristics

TL;DR

This paper introduces universal families of Eulerian multiple zeta values in positive characteristic, generalizing Thakur's values for curves over finite fields and proving a related conjecture.

Contribution

It establishes a non-commutative factorization of exponential series, introduces universal families of multiple zeta values, and proves a conjecture by Lara Rodriguez and Thakur.

Findings

Universal families of Eulerian multiple zeta values are established.

A general non-commutative factorization of exponential series is proven.

The conjecture of Lara Rodriguez and Thakur is confirmed.

Abstract

We study positive characteristic multiple zeta values associated to general curves over $\mathbb F_q$ together with an $\mathbb F_q$-rational point $\infty$ as introduced by Thakur. For the case of the projective line these values were defined as analogues of classical multiple zeta values. In the present paper we first establish a general non-commutative factorization of exponential series associated to certain lattices of rank one. Next we introduce universal families of multiple zeta values of Thakur and show that they are Eulerian in full generality. In particular, we prove a conjecture of Lara Rodriguez and Thakur arXiv:2003.12910. One of the main ingredients of the proofs is the notion of L-series in Tate algebras introduced by the third author arXiv:1107.4511 in 2012.