English

Symmetric multiple Eisenstein series

Number Theory 2026-01-21 v1

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 kk. When kk is even, it coincides with the space spanned by modular forms of weight kk and the derivative of the Eisenstein series of weight k2k-2. For kk odd, we prove that its dimension equals k/3\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.

Keywords

Cite

@article{arxiv.2601.13626,
  title  = {Symmetric multiple Eisenstein series},
  author = {Takashi Hara and Kenji Sakugawa and Koji Tasaka},
  journal= {arXiv preprint arXiv:2601.13626},
  year   = {2026}
}

Comments

43 pages

R2 v1 2026-07-01T09:11:53.091Z