Arithmetic properties of generalized hecke operators

다국어 초록 (Multilingual Abstract)

The concept of the generalized Hecke operator originates from the replication formulae in Monstrous Moonshine. The main objective of this dissertation is to extend the construction of generalized Hecke operators to meromorphic modular forms, with a primary focus on the case of weakly holomorphic modular forms, and to investigate their arithmetic properties. When the level is $1$, the generalized Hecke operator coincides with the classical Hecke operator; for a general level $N$, it also agrees with the classical one when the index and the level are coprime. The distinction arises precisely when they are not coprime. In classical Hecke theory, many results are formulated under the coprimality assumption. The generalized Hecke operator theory naturally extends these results and complements several classical phenomena, including Zagier duality, Hecke equivariance on generating functions, and modular equations. In addition, we present an approach to constructing generalized Hecke operators through an appropriate Hecke algebra.

목차 (Table of Contents)

• 1. Introduction 1
• 2. Preliminaries 17
• 2.1 Modular forms 17
• 2.2 Hecke operators 21
• 2.2.1 Hecke operators via lattices 21
• 2.2.2 Hecke operators via Hecke algebra 25
• 2.3 Bases for weakly holomorphic modular forms 28
• 3. Generalized Hecke operators 33
• 3.1 Generalized Hecke operators via lattices 33
• 3.2 Generalized Hecke operators via Hecke algebra 37
• 3.3 Algebraic structure for generalized Hecke operators 40
• 4. Generalized Hecke operators on weakly holomorphic modular forms 51
• 4.1 Generalized Hecke operators on weakly holomorphic modular forms 51
• 4.2 Zagier duality under generalized Hecke operators 54
• 4.3 Hecke equivariance on the generating function 58
• 4.4 generalized modular equation 63
• 5. Outlook 76