Quantum probabilistic study of parametrized trace formulas and L-functions

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14298
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Kanazawa University (2023); Nihon University (2020-2022)
• Principal Investigator: Sugiyama Shingo 金沢大学, 数物科学系, 准教授 (70821817)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 保型L関数 / 保型形式 / ディリクレL関数 / 零点分布 / 1レベル密度 / 特殊値 / 重み付き零点分布 / ランダム行列

Outline of Final Research Achievements

By a parametrized trace formula, we found a new phenomenon on the density of zeros of symmetric power L-functions weighted by L-values. The phenomenon is that the density is changed if and only if the weight factors are central L-values. Based on this, we suggested a conjecture on the weighted density of zeros of L-functions in a family. Furthermore, we confirmed the conjecture for a family of Dirichlet L-functions, which is a joint work with Ade Irma Suriajaya (Kyushu University). We completed to write a paper on the integrality of Hecke eigenvalues and its applications to Hecke fields. (The content of the paper itself was already completed essentially before.) This is a joint work with Kenji Sakugawa (Shinshu University).

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

本研究では, L関数の零点分布はランダム行列の固有値分布と同じであろうというKatz-Sarnakの予想の重み付き版を考察した. L関数の特殊値による重み付き零点分布を考察することで, L関数の中心値の特異性を見出す契機となった. 従来のL関数の零点分布は, 重み付き零点分布であっても密度関数がランダム行列理論に出てくる5種類のいずれかに集約されていた. しかし本研究ではその5種類とは異なる密度関数を2種類発見した. またHecke固有値の代数的整数性は一般のHilbert保型形式と一般のSiegel保型形式の場合を扱っているため, 論文が完成し皆が閲覧できる状態になったことには意義がある.