Modular Forms with Integral and Half Integral Weights Modular Forms with Integral and Half Integral Weights focuses on the fundamental theory of modular forms of one variable with integral and half integral weights. The main theme of the book is the theory of Eisenstein series. It is a fundamental problem to construct a basis of the orthogonal complement of the space of cusp forms; as is well known this space is spanned by Eisenstein series for any weight greater than or equal to 2. The book proves that the conclusion holds true for weight 3 2 by explicitly constructing a basis of the orthogonal complement of the space of cusp forms. The problem for weight 1 2 which was solved by Serre and Stark will also be discussed in this book. The book provides readers not only basic knowledge on this topic but also a general survey of modern investigation methods of modular forms with integral and half integral weights. It will be of significant interest to researchers and practitioners in modular forms of mathematics. Dr. Xueli Wang is a Professor at South China Normal University China. Dingyi Pei is a Professor at Guangzhou University China. Theta Functions and Their Transformation Formulae Eisenstein Series The Modular Group and Its Subgroups Modular Forms with Integral Weight or Half integral Weight Operators on the Space of Modular Forms New Forms and Old Forms. Construction of Eisenstein Series Weil Representation and Shimura Lifting Trace Formula Integers Represented by Positive Definite Quadratic Forms.