分割數函數, 志村對應, 四元theta函數及 L-函數的特殊值

Abstract

在本計畫中，我們將研究數個有關模形式的問題。包含 1. 分割數函數及其推廣的同餘性質， 2. 權為整數的模形式及權為半整數的模形式間的志村對應， 3. Hecke算子在Atkin-Lehner特徵子空間的trace公式及其應用， 4. 四元數的theta函數與模形式的關聯， 5. 四元數的theta函數的線性關係與橢圓曲線的算術性質的關聯。

In this research project, we will study several problems related to the theory of modular forms, including 1. Congruences of the partition function and its generalization, 2. Shimura correspondence between certain modular forms of half-integral weights and modular forms of integral weights. 3. Trace formulas for Hecke operators on Atkin-Lehner eigensubspaces of cusp forms. 4. Theta series attached to quaternion orders over Q and their relations to modular forms. 5. Linear dependences among theta series attached to quaternion orders over Q and their relations to arithmetic properties of elliptic curves over Q.