Symmetric multiple Eisenstein series

TL;DR

This paper introduces symmetric multiple Eisenstein series, explores their algebraic relations, and connects them to modular forms and elliptic double zeta values, advancing understanding of their structure and modular properties.

Contribution

It defines symmetric multiple Eisenstein series, establishes their shuffle relations, and links them to modular forms and elliptic double zeta values, providing new insights into their algebraic and modular structures.

Findings

Symmetric multiple Eisenstein series satisfy linear shuffle relations.

For even weight k, the space spanned matches modular forms and derivatives.

For odd weight k, the dimension is approximately k/3.

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.