Algebra Structures of Multiple Eisenstein Series in Positive Characteristic

TL;DR

This paper investigates the algebraic structure of multiple Eisenstein series in positive characteristic, proving their linear independence, embedding properties, and associativity, thus confirming a prior conjecture.

Contribution

It establishes the algebraic independence, embedding of multiple zeta values, and the isomorphism of the $q$-shuffle algebra, confirming conjectures about their structure.

Findings

Proved linear independence of multiple Eisenstein series.

Embedded the $q$-shuffle algebra of multiple zeta values into inverse limits.

Showed the algebra $ ext{E}$ is associative, confirming the conjecture.

Abstract

In [CCHT25], the authors introduced multiple Eisenstein series of arbitrary rank in positive characteristic and the $q$-shuffle algebra $\mathcal{E}$ associated with them. In the present paper, we establish a class of linear independence results for multiple Eisenstein series. We also prove that the $q$-shuffle algebra $\mathcal{R}$ of multiple zeta values embeds into the inverse limit of the spaces of multiple Eisenstein series with respect to the rank $r$, and that $\mathcal{E}$ is isomorphic to the tensor square of $\mathcal{R}$. As an application, we show that $\mathcal{E}$ is an associative algebra, thereby verifying the conjecture proposed in [CCHT25]