Elementary Modular Iwasawa Theory

This comprehensive introduction to Iwasawa theory blends modular forms and Galois representations, making complex ideas accessible to researchers and students. Elementary Modular Iwasawa Theory encourages exploration.

This book serves as a pioneering work that offers a thorough and accessible introduction to the innovative Iwasawa theory, which has been developed through the lens of deformation theory of modular forms and Galois representations. The author expands on the cohomological approach introduced in their award-winning 2019 AMS Leroy P Steele Prize article, steering clear of excessive reliance on algebraic geometry. By doing so, this text makes complex concepts more approachable for readers.

Beginning with an exploration of Iwasawa's classical findings, the book delves into his proof of the main conjecture under the Kummer-Vandiver conjecture. This foundational work not only establishes the cyclicity of Iwasawa's module but also extends the understanding of his main conjecture. The author then presents a generalization of the methods used to prove cyclicity, applying them to the adjoint Selmer group of various ordinary deformations of two-dimensional Artin Galois representations.

In the initial chapters, the book lays out essential principles and introduces numerous open problems designed to inspire and engage young researchers in their academic pursuits. By addressing these challenges, Elementary Modular Iwasawa Theory aims to foster a deeper understanding of this vital area of study and encourage further exploration within the field.