This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the Shimura-Taniyama conjecture, is given. In addition, the book presents an outline of the proof of diverse modularity results of two-dimensional Galois representations (including that of Wiles), as well as some of the author's new results in that direction.Contents:An Algebro-Geometric Tool BoxElliptic CurvesGeometric Modular FormsJacobians and Galois RepresentationsModularity ProblemsReadership: Graduates and researchers in number theory.Key Features:The author is the former Chief Economist and Senior Vice President of Development Economics at the World BankThe author, who is one of China's leading economists, has a profound understanding of China's social, economic and political structuresThis book not only investigates problems and challenges faced by the Chinese economy, but also sheds new light on the solutions and opportunities