| 研究課題/領域番号 |
21K13771
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| 研究機関 | 名古屋大学 |
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研究代表者 |
BACHMANN Henrik 名古屋大学, 多元数理科学研究科, 特任助教 (20813372)
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| 研究期間 (年度) |
2021-04-01 – 2023-03-31
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| キーワード | multiple zeta values / Eisenstein series / q-analogues of MZV / Kronecker function / modular forms |
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| 研究実績の概要 |
In a joint work with A. Burmester we finished a preprint on "Combinatorial multiple Eisenstein series". In this work we introduce a generalization of the extended double shuffle relations and give a solution to these in terms of formal power series with rational coefficients. The construction is inspired by the classical calculation of the Fourier expansion of multiple Eisenstein series, but needs some extra ingredients. As an application one obtains purely combinatorial proofs of relations among modular forms. In another project, joint with U. Kuehn and N. Matthes, we give another definition of combinatorial multiple Eisenstein series in depth two. This is done by generalizing a construction of Gangl-Kaneko-Zagier for rational solutions to the double shuffle relations in depth two. In our project we show that power series satisfying the so-called Fay idenity can be used to obtain solutions to the generalized double shuffle relations in depth two. Applying this construction to the Kronecker function then yields a definition of double Eisenstein series.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
The current research plan is going as expected. The work on combinatorial multiple Eisenstein series gave a lot of new open questions for further projects.
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| 今後の研究の推進方策 |
Currently in a joint project with J.-W. van Ittersum and N. Matthes we are investigating formal multiple Eisenstein series. These can be seen as a formal analogue of the combinatorial multiple Eisenstein series. In this work we give the algebraic describtion of generalized double shuffle relations and we show how these are related to the classical extended double shuffle relations of multiple zeta values. Based on computer based experiments we also have a conjectured sl_2 action on our space, which seems to be a natural extension of the usual sl_2 action on quasi-modular forms. The proof of this conjectured action is still work in progress but also seems to be in reach.
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| 次年度使用額が生じた理由 |
For the project on formal multiple Eisenstein series the plan is to travel to Germany in August to visit the University of Hamburg, give a presentation on a conference in Bielefeld and to visit the MPIM in Bonn. Further I would like to invite on of my collaborators to Japan at the end of this year or the beginning of next year. Besides this the grant will be used for travel around Japan for attending conferences and giving talks.
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