2022 Fiscal Year Annual Research Report
Generalizations of the double shuffle relations for multiple zeta values and the connections to modular forms
| Project/Area Number |
21K13771
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| Research Institution | Nagoya University |
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Principal Investigator |
BACHMANN Henrik 名古屋大学, 多元数理科学研究科(国際), G30特任准教授 (20813372)
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| Project Period (FY) |
2021-04-01 – 2023-03-31
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| Keywords | multiple zeta values / Eisenstein series / modular forms |
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| Outline of Annual Research Achievements |
In this research project a generalization of the classical double shuffle relations of multiple zeta values were introduced and studied. This new set of equations are motivated by multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. It is defined by using the notion of bimoulds and they are given by those bimoulds which are symmetril and swap invariant. Coefficients of bimoulds satisfying this property are called formal multiple Eisenstein series. We show (j.w. A. Burmester) that the space of formal multiple Eisenstein series has a realization given by an algebra homomorphism into the space of formal q-series with rational coefficients. In depth one these are exactly Eisenstein series and their derivatives. This gives a natural bridge between the theory of modular forms and multiple zeta values. In another project (j.w. J.W. van Ittersum) we show that the space of formal multiple Eisenstein series is an sl2-algebra. This generalized the sl2-modular structure of quasimodular forms and gives new insights into the study of multiple Eisenstein series and q-analogues of multiple zeta values. Another result of this study is a new notion of formal (quasi)modular forms equipped with a formal version of the natural derivation qd/dq.
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