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Algebra Structures of Multiple Eisenstein Series in Positive Characteristic

Ting-Wei Chang, Song-Yun Chen, Fei-Jun Huang, Hung-Chun Tsui

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]

Algebra Structures of Multiple Eisenstein Series in Positive Characteristic

Abstract

In [CCHT25], the authors introduced multiple Eisenstein series of arbitrary rank in positive characteristic and the -shuffle algebra associated with them. In the present paper, we establish a class of linear independence results for multiple Eisenstein series. We also prove that the -shuffle algebra of multiple zeta values embeds into the inverse limit of the spaces of multiple Eisenstein series with respect to the rank , and that is isomorphic to the tensor square of . As an application, we show that is an associative algebra, thereby verifying the conjecture proposed in [CCHT25]
Paper Structure (10 sections, 21 theorems, 117 equations)

This paper contains 10 sections, 21 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. The q-shuffle relation for multiple Eisenstein series
  3. Main results
  4. t-expansion of multiple Eisenstein series and applications
  5. Multiple Goss sums and Goss expansions
  6. t-expansion of multiple Goss sums and multiple Eisenstein series
  7. Linear independence of multiple Eisenstein series
  8. Algebra structure on the q-shuffle algebra E
  9. Associativity of E
  10. Further discussion of E

Key Result

Theorem 1.4

For $r\ge 1$, we define $\widehat{E}_r: \mathcal{R} \to \mathcal{O}(\Omega^r)$ to be the unique $\mathbb{F}_p$-linear map satisfying Then $\widehat{E}_r$ is an $\mathbb{F}_p$-algebra homomorphism, i.e.,

Theorems & Definitions (53)

  • Definition 1.1
  • Definition 1.2: Multiple Eisenstein series
  • Definition 1.3
  • Theorem 1.4: CCHT2025
  • Definition 1.5
  • Theorem 1.6: restated as Proposition \ref{['prop-taking-constant-map']} and Theorem \ref{['thm-Rcal-inverse-limit']}
  • Corollary 1.7: restated as Corollary \ref{['cor-assoc-of-R']}
  • Remark 1.8
  • Definition 1.9
  • Remark 1.10
  • ...and 43 more