Symmetric multiple Eisenstein series
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866909995571871744 |
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| author | Hara, Takashi Sakugawa, Kenji Tasaka, Koji |
| author_facet | Hara, Takashi Sakugawa, Kenji Tasaka, Koji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13626 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetric multiple Eisenstein series Hara, Takashi Sakugawa, Kenji Tasaka, Koji Number Theory 11M32, 11F11 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. |
| title | Symmetric multiple Eisenstein series |
| topic | Number Theory 11M32, 11F11 |
| url | https://arxiv.org/abs/2601.13626 |