This work offers an in-depth exploration of the latest advancements in the arithmetic theory of modular forms, making it an essential resource for graduate students and researchers in the field. It delves into complex concepts, providing a thorough understanding of the subject's evolution and current trends. The book emphasizes both theoretical frameworks and practical applications, catering to those seeking to deepen their knowledge and research in modular forms.
The book delves into the theory of moduli spaces of elliptic curves and their applications to modular forms, detailing the construction of Galois representations crucial to Wiles' proof of the Shimura Taniyama conjecture. The second edition enhances the content with a thorough exploration of Barsotti Tate groups and includes a practical overview of formal deformation theory of elliptic curves. Additionally, it presents new results on Ribet's theorem and a glimpse into contemporary research in Number Theory, particularly regarding modularity theory of abelian varieties and curves.
The book offers a groundbreaking exploration of Iwasawa theory through the lens of deformation theory of modular forms and Galois representations. It expands upon the cohomological methods established in a previous award-winning article, presenting concepts in an accessible manner while minimizing reliance on algebraic geometry. This comprehensive account aims to make advanced theories more approachable for readers, enhancing understanding of the intricate relationships between modular forms and Galois representations.
Focusing on Shimura varieties, this book delves into significant topics such as arithmetic invariants, special values of L-functions, and the study of elliptic curves across complex and p-adic fields. It also explores Hecke algebras and the intricate relationships between elliptic and modular curves over rings, providing a comprehensive introduction to these advanced concepts in algebraic geometry and number theory.
This book covers the following three topics in a manner accessible to graduate students who have an understanding of algebraic number theory and scheme theoretic algebraic geometry: 1. An elementary construction of Shimura varieties as moduli of abelian schemes. 2. p-adic deformation theory of automorphic forms on Shimura varieties. 3. A simple proof of irreducibility of the generalized Igusa tower over the Shimura variety. The book starts with a detailed study of elliptic and Hilbert modular forms and reaches to the forefront of research of Shimura varieties associated with general classical groups. The method of constructing p-adic analytic families and the proof of irreducibility was recently discovered by the author. The area covered in this book is now a focal point of research worldwide with many far-reaching applications that have led to solutions of longstanding problems and conjectures. Specifically, the use of p-adic elliptic and Hilbert modular forms have proven essential in recent breakthroughs in number theory (for example, the proof of Fermat's Last Theorem and the Shimura-Taniyama conjecture by A. Wiles and others). Haruzo Hida is Professor of Mathematics at University of California, Los Angeles. His previous books include Modular Forms and Galois Cohomology (Cambridge University Press 2000) and Geometric Modular Forms and Elliptic Curves (World Scientific Publishing Company 2000).
