Asymptotic formula of Hecke eigenvalues and research of Arthur trace formula

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03235
• 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: Kanazawa University
• Principal Investigator: Wakatsuki Satoshi 金沢大学, 数物科学系, 教授 (10432121)
• Co-Investigator(Kenkyū-buntansha): 都築 正男 上智大学, 理工学部, 教授 (80296946)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 数論 / 保型形式 / 跡公式 / ヘッケ固有値

Outline of Final Research Achievements

In this research, we studied Hecke eigenvalues of automorphic forms in number theory. Number theory is a field that studies various properties of natural numbers and primes, automorphic forms are mysterious functions which have high symmetry, and their Hecke eigenvalues are sequences of numbers which naturally arise from them. Hecke eigenvalues have very interesting number-theoretic properties, and a classical example of Hecke eigenvalues is Ramanujan's tau function. A major result of our research is that we have succeeded to prove various results on the distributions of their eigenvalues by considering natural families of automorphic forms.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

保型形式の族のヘッケ固有値の分布はプランシュレル測度や佐藤-テイト測度に従うことが予想されており、様々な場合に証明されている。ヘッケ固有値の漸近公式の一般化と精密化を推進することで、ヘッケ固有値の分布の性質をより統一的に明らかにすることが本研究の目的であった。実際、本研究の成果によって、その一般化と精密化の両方についてヘッケ固有値の漸近公式の研究を大きく進展させることができた。