Association of multiple modular L-functions and polylogarithms

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03260
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Kindai University
• Principal Investigator: Ihara Kentaro 近畿大学, 理工学部, 准教授 (00467523)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: multiple L function / multiple zeta values / elliptic modular form / harmonic product / Mellin transformation / Hurwitz zeta function / Dedekind zeta function

Outline of Final Research Achievements

One of the natural question on the period algebra associated to multiple automorphic L-values is whether the algebra is equipped with the harmonic product, including the connection with Rankin convolution. In collaboration with Dr. Nakamura of Kindai University and Dr. Yamamoto of Keio University, we obtained the Mellin integral representation of multiple zeta values. It turns out that the harmonic product originates from the product of its integrands. As a shuffle version of this research, in collaboration with Mr. Ishibashi of Kindai University, we obtained the Mellin integral representation of multiple Hurwitz polylogs. In addition, through research with Dr. Nakamura and others at Kinki University, we obtained the Ohno-Zagier type relations for multiple Hurwitz zeta values, and this result was published in a journal in 2022. As a recent result with Mr. Matsuda of Kindai University, we found a relationship between automorphic multiple L-values and Dedekind zeta values.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

保型形式と呼ばれる関数に対して、周期という積分を用いて表示される特別な数が定まる。本研究では、従来の周期の多重化を考察した。様々な保型形式を考え、その多重化された周期の全体に備わる構造の研究を行った結果、保型形式以外の整数論に現れる種々のゼータ関数やL関数、またポリログ関数の多重化との関係をいくつか見つけることができた。例えば、保型形式にも関連するポリログ関数やフルヴィッツ型のポリログ関数のメリン型積分による表示を発見することができた。