Study of modular/quasimodular forms and multiple zeta values appearing in various aspects of mathematics and physics

2015 Fiscal Year Final Research Report

• Project/Area Number: 23340010
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: Kaneko Masanobu 九州大学, 数理(科)学研究科(研究院), 教授 (70202017)
• Co-Investigator(Renkei-kenkyūsha): MATSUMOTO Makoto 広島大学, 理学研究科, 教授 (70231602) ; MURAKAMI Jun 早稲田大学, 理工学術院, 教授 (90157751) ; NAGATOMO Kiyokazu 大阪大学, 情報科学研究科, 准教授 (90172543) ; HOSONO Shinobu 東京大学, 数理科学研究科, 准教授 (60212198) ; HIKAMI Kazuhiro 九州大学, 数理学研究院, 准教授 (60262151) ; TAGUCHI Yuichiro 東京工業大学, 理工学研究科, 教授 (90231399) ; TAKATA Toshie 九州大学, 数理学研究院, 准教授 (40253398)
• Project Period (FY): 2011-04-01 – 2016-03-31
• Keywords: モジュラー形式 / 準モジュラー形式 / 多重ゼータ値 / 共系場理論 / 位相不変量

Outline of Final Research Achievements

For modular forms of one variable, we obtained some congruence results on Fourier coefficients of certain　meromorphic modular forms, constructed newforms associated to elliptic curves over the rationals which can be written as eta products via differential equations satisfied by　modular forms, and found relations between period polynomials of modular forms and double zeta values of level 2. This last result creates a bridge between modular forms and multiple zeta values. For multiple zeta values proper, we obtained a formula for the height one MZV in terms of MZVs of maximal height, defined and studied a kind of sibling function of the so called Arakawa-Kaneko zeta function, and studied basic properties of finite multiple zeta values. Notably, by introducing symmetric multiple zeta values, we proposed an astonishing main conjecture in the theory of finite multiple zeta values.

Free Research Field

代数学