by Haruzo Hida (University of California, Los Angeles)

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.

• An Algebro-Geometric Tool Box
• Elliptic Curves
• Geometric Modular Forms
• Jacobians and Galois Representations
• Modularity Problems