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Softcover ISBN:  9781470456726 
Product Code:  SURV/272 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470473259 
Product Code:  SURV/272.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470456726 
eBook ISBN:  9781470473259 
Product Code:  SURV/272.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 272; 2023; 154 ppMSC: Primary 11; 13;
Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to \(p\)adic \(L\)functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation of this book is an update of the classical theory for class groups taking into account the changed point of view on Iwasawa theory.
The goal of this first part of the twopart publication is to explain the theory of ideal class groups, including its algebraic aspect (the Iwasawa class number formula), its analytic aspect (Leopoldt–Kubota \(L\)functions), and the Iwasawa main conjecture, which is a bridge between the algebraic and the analytic aspects.
The second part of the book will be published as a separate volume in the same series, Mathematical Surveys and Monographs of the American Mathematical Society.
ReadershipGraduate students and researchers interested in number theory and arithmetic geometry.

Table of Contents

Chapters

Motivation and utility of Iwasawa theory

$\mathbb {Z}_p$extension and Iwasawa algebra

Cyclotomic Iwasawa theory for ideal class groups

Bookguide

Appendix A


Additional Material

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Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to \(p\)adic \(L\)functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation of this book is an update of the classical theory for class groups taking into account the changed point of view on Iwasawa theory.
The goal of this first part of the twopart publication is to explain the theory of ideal class groups, including its algebraic aspect (the Iwasawa class number formula), its analytic aspect (Leopoldt–Kubota \(L\)functions), and the Iwasawa main conjecture, which is a bridge between the algebraic and the analytic aspects.
The second part of the book will be published as a separate volume in the same series, Mathematical Surveys and Monographs of the American Mathematical Society.
Graduate students and researchers interested in number theory and arithmetic geometry.

Chapters

Motivation and utility of Iwasawa theory

$\mathbb {Z}_p$extension and Iwasawa algebra

Cyclotomic Iwasawa theory for ideal class groups

Bookguide

Appendix A