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Symmetric multiple Eisenstein series

Takashi Hara, Kenji Sakugawa, Koji Tasaka

TL;DR

This work introduces symmetric multiple Eisenstein series $G^{\shuffle,S}_{\boldsymbol{k}}(\tau)$ as a shuffle-regularized analogue of classical multiple Eisenstein series, linking their constant terms to symmetric multiple zeta values. The authors prove that these symmetric series satisfy shuffle relations and establish deep connections to modular forms, elliptic zeta values, and Fay-shuffle spaces. They determine the structure and dimensions of the spaces spanned by symmetric double Eisenstein series at even and odd weights, and show that modular forms can be expressed via symmetric triple Eisenstein series, providing a first step toward modular phenomena for symmetric zeta values. Representation-theoretic and combinatorial tools, including $\mathfrak{S}_3$-representations and the linear shuffle space, underpin the dimension counts and dualities with period polynomials, completing a comprehensive framework linking symmetric MZVs, modular forms, and Eisenstein-series regularizations.

Abstract

In this paper, we introduce the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relation. As a case study, we investigate the vector space spanned by symmetric double Eisenstein series of weight $k$. When $k$ is even, it coincides with the space spanned by modular forms of weight $k$ and the derivative of the Eisenstein series of weight $k-2$. For $k$ odd, we prove that its dimension equals $\lfloor k/3\rfloor$. We further provide an explicit correspondence between the linear shuffle relation and the Fay-shuffle relation satisfied by elliptic double zeta values, which may be of independent interest. In connection with modular forms, we prove that every modular form can be expressed as a linear combination of symmetric triple Eisenstein series. This will serve as a first step toward understanding modular phenomena for symmeric multiple zeta values observed by Kaneko and Zagier.

Symmetric multiple Eisenstein series

TL;DR

This work introduces symmetric multiple Eisenstein series as a shuffle-regularized analogue of classical multiple Eisenstein series, linking their constant terms to symmetric multiple zeta values. The authors prove that these symmetric series satisfy shuffle relations and establish deep connections to modular forms, elliptic zeta values, and Fay-shuffle spaces. They determine the structure and dimensions of the spaces spanned by symmetric double Eisenstein series at even and odd weights, and show that modular forms can be expressed via symmetric triple Eisenstein series, providing a first step toward modular phenomena for symmetric zeta values. Representation-theoretic and combinatorial tools, including -representations and the linear shuffle space, underpin the dimension counts and dualities with period polynomials, completing a comprehensive framework linking symmetric MZVs, modular forms, and Eisenstein-series regularizations.

Abstract

In this paper, we introduce the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relation. As a case study, we investigate the vector space spanned by symmetric double Eisenstein series of weight . When is even, it coincides with the space spanned by modular forms of weight and the derivative of the Eisenstein series of weight . For odd, we prove that its dimension equals . We further provide an explicit correspondence between the linear shuffle relation and the Fay-shuffle relation satisfied by elliptic double zeta values, which may be of independent interest. In connection with modular forms, we prove that every modular form can be expressed as a linear combination of symmetric triple Eisenstein series. This will serve as a first step toward understanding modular phenomena for symmeric multiple zeta values observed by Kaneko and Zagier.
Paper Structure (26 sections, 29 theorems, 184 equations, 6 tables)

This paper contains 26 sections, 29 theorems, 184 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the article
  4. Multiple Eisenstein series
  5. Algebraic setups on multiple zeta values
  6. Multiple Eisenstein series and its Fourier expansion
  7. Shuffle regularization of multiple Eisenstein series
  8. Alternative expression using the Ihara group law
  9. Comparison between linear relations of $G^\shuffle_{{\boldsymbol{k}}}$ and $\tilde{g}^\shuffle_{{\boldsymbol{k}}}$
  10. Symmetric multiple Eisenstein series
  11. Review on symmetric multiple zeta values
  12. Symmetric multiple Eisenstein series and its basic properties
  13. Numerical observations
  14. Linear shuffle space
  15. Power series expression of the shuffle relation
  16. ...and 11 more sections

Key Result

Theorem 1.1

Every cusp form of weight $k$ for ${\rm SL}_2({\mathbb Z})$ with rational Fourier coefficients is presented as a ${\mathbb Q}$-linear combination of symmetric triple Eisenstein series $\widetilde{G}_{k_1,k_2,k_3}^{\shuffle,S}(\tau)$ for $k_1,k_2,k_3\in {\mathbb N}$ with $k=k_1+k_2+k_3$.

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: =Theprem \ref{['thm:lin_sh_vs_fay']}
  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 53 more