q-analogues of multiple zeta values and their applications in geometry

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14499
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Nagoya University
• 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)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: Multiple zeta values / Modular forms / Functions on partitions

Outline of Final Research Achievements

In the project "q-analogues of multiple zeta values and their applications in geometry" the connection of q-analogues and the study of a more broader class of q-series were studied. For this we (j.w. with Jan-Willem van Ittersum) introduced the notion of polynomial functions on partitions. The main result is that all these functions, which are given by the q-bracket of certain polynomials, are always give rise to qanalogues of multiple zeta values. In particular, we calculated the limit as q goes to 1. As an application we showed how these connections give rise to relations among multiple zeta values. In another project (j.w. Ulf Kuehn and Nils Matthes) we introduced the notion of the formal double Eisenstein space. This space can be seen as a generalization of the formal double zeta space introduced by Gangl-Kaneko-Zagier. We showed that any power series satisfying the Fay-idendity give rise to a realization of this space.

Free Research Field

Number theory

Academic Significance and Societal Importance of the Research Achievements

The introduction of the theory of polynomial functions on partitions builds a new bridge between the theory of partitions and multiple zeta values. This gives for example new families of relations among multiple zeta values coming from the theory of modular forms.