Iwasawa Theory and Its Perspective, Volume 2
Iwasawa Theory and Its Perspective, Volume 2

Author: Tadashi Ochiai

Publisher: American Mathematical Society

Published: 2024-04-25

Total Pages: 228

ISBN-13: 1470456737

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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 is to update the classical theory for class groups, taking into account the changed point of view on Iwasawa theory. The goal of this second part of the three-part publication is to explain various aspects of the cyclotomic Iwasawa theory of $p$-adic Galois representations.

Iwasawa Theory and Its Perspective, Volume 1
Iwasawa Theory and Its Perspective, Volume 1

Author: Tadashi Ochiai

Publisher: American Mathematical Society

Published: 2023-05-03

Total Pages: 167

ISBN-13: 1470456729

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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 two-part 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.

Automorphic Forms Beyond $mathrm {GL}_2$
Automorphic Forms Beyond $mathrm {GL}_2$

Author: Ellen Elizabeth Eischen

Publisher: American Mathematical Society

Published: 2024-03-26

Total Pages: 199

ISBN-13: 1470474921

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The Langlands program has been a very active and central field in mathematics ever since its conception over 50 years ago. It connects number theory, representation theory and arithmetic geometry, and other fields in a profound way. There are nevertheless very few expository accounts beyond the GL(2) case. This book features expository accounts of several topics on automorphic forms on higher rank groups, including rationality questions on unitary group, theta lifts and their applications to Arthur's conjectures, quaternionic modular forms, and automorphic forms over functions fields and their applications to inverse Galois problems. It is based on the lecture notes prepared for the twenty-fifth Arizona Winter School on “Automorphic Forms beyond GL(2)”, held March 5–9, 2022, at the University of Arizona in Tucson. The speakers were Ellen Eischen, Wee Teck Gan, Aaron Pollack, and Zhiwei Yun. The exposition of the book is in a style accessible to students entering the field. Advanced graduate students as well as researchers will find this a valuable introduction to various important and very active research areas.

Elementary Modular Iwasawa Theory
Elementary Modular Iwasawa Theory

Author: Haruzo Hida

Publisher: World Scientific

Published: 2021-10-04

Total Pages: 446

ISBN-13: 9811241384

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This book is the first to provide a comprehensive and elementary account of the new Iwasawa theory innovated via the deformation theory of modular forms and Galois representations. The deformation theory of modular forms is developed by generalizing the cohomological approach discovered in the author's 2019 AMS Leroy P Steele Prize-winning article without using much algebraic geometry.Starting with a description of Iwasawa's classical results on his proof of the main conjecture under the Kummer-Vandiver conjecture (which proves cyclicity of his Iwasawa module more than just proving his main conjecture), we describe a generalization of the method proving cyclicity to the adjoint Selmer group of every ordinary deformation of a two-dimensional Artin Galois representation.The fundamentals in the first five chapters are as follows:Many open problems are presented to stimulate young researchers pursuing their field of study.

Iwasawa Theory 2012
Iwasawa Theory 2012

Author: Thanasis Bouganis

Publisher: Springer

Published: 2014-12-08

Total Pages: 487

ISBN-13: 3642552455

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This is the fifth conference in a bi-annual series, following conferences in Besancon, Limoges, Irsee and Toronto. The meeting aims to bring together different strands of research in and closely related to the area of Iwasawa theory. During the week before the conference in a kind of summer school a series of preparatory lectures for young mathematicians was provided as an introduction to Iwasawa theory. Iwasawa theory is a modern and powerful branch of number theory and can be traced back to the Japanese mathematician Kenkichi Iwasawa, who introduced the systematic study of Z_p-extensions and p-adic L-functions, concentrating on the case of ideal class groups. Later this would be generalized to elliptic curves. Over the last few decades considerable progress has been made in automorphic Iwasawa theory, e.g. the proof of the Main Conjecture for GL(2) by Kato and Skinner & Urban. Techniques such as Hida’s theory of p-adic modular forms and big Galois representations play a crucial part. Also a noncommutative Iwasawa theory of arbitrary p-adic Lie extensions has been developed. This volume aims to present a snapshot of the state of art of Iwasawa theory as of 2012. In particular it offers an introduction to Iwasawa theory (based on a preparatory course by Chris Wuthrich) and a survey of the proof of Skinner & Urban (based on a lecture course by Xin Wan).

Number Theory
Number Theory

Author: Kazuya Kato

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 243

ISBN-13: 0821820958

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Number Theory 1
Number Theory 1

Author: Kazuya Kato

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 180

ISBN-13: 9780821808634

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This is the English translation of the original Japanese book. In this volume, "Fermat's Dream", core theories in modern number theory are introduced. Developments are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the number fields. This work presents an elegant perspective on the wonder of numbers. Number Theory 2 on class field theory, and Number Theory 3 on Iwasawa theory and the theory of modular forms, are forthcoming in the series.

Number Theory 3
Number Theory 3

Author: Nobushige Kurokawa

Publisher:

Published: 2012

Total Pages: 242

ISBN-13: 9780821891629

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This is the third of three related volumes on number theory. (The first two volumes were also published in the Iwanami Series in Modern Mathematics, as volumes 186 and 240.) The two main topics of this book are Iwasawa theory and modular forms. The presentation of the theory of modular forms starts with several beautiful relations discovered by Ramanujan and leads to a discussion of several important ingredients, including the zeta-regularized products, Kronecker's limit formula, and the Selberg trace formula. The presentation of Iwasawa theory focuses on the Iwasawa main conjecture, which est.

Introduction to $p$-adic Analytic Number Theory
Introduction to $p$-adic Analytic Number Theory

Author: M. Ram Murty

Publisher: American Mathematical Soc.

Published: 2009-02-09

Total Pages: 162

ISBN-13: 0821847740

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This book is an elementary introduction to $p$-adic analysis from the number theory perspective. With over 100 exercises included, it will acquaint the non-expert to the basic ideas of the theory and encourage the novice to enter this fertile field of research. The main focus of the book is the study of $p$-adic $L$-functions and their analytic properties. It begins with a basic introduction to Bernoulli numbers and continues with establishing the Kummer congruences. These congruences are then used to construct the $p$-adic analog of the Riemann zeta function and $p$-adic analogs of Dirichlet's $L$-functions. Featured is a chapter on how to apply the theory of Newton polygons to determine Galois groups of polynomials over the rational number field. As motivation for further study, the final chapter introduces Iwasawa theory.

Working Minimalism
Working Minimalism

Author: Samuel David Epstein

Publisher: MIT Press

Published: 1999

Total Pages: 380

ISBN-13: 9780262550321

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Essays present explicit syntactic analyses that adhere to programmatic minimalist guidelines. The essays in this book present explicit syntactic analyses that adhere to programmatic minimalist guidelines. Thus they show how the guiding ideas of minimalism can shape the construction of a new, more explanatory theory of the syntactic component of the human language faculty. Contributors Zeljko Boskovic, Samuel David Epstein, Robert Freidin, Erich M. Groat, Norbert Hornstein, Hisatsugu Kitahara, Howard Lasnik, Roger Martin, Jairo Nunes, Norvin Richards, Juan Uriagereka, Amy Weinberg Current Studies in Linguistics No. 32