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Number Theory, Volume 1,2,3 (Iwanami Series in Modern Mathematics)
Kazuya Kato | 2000-2012 | ISBN: 082180863X; 0821813552; 0821820958 | English | 174+258+242 pages | PDF | 19 MB
Series: Translations of Mathematical Monographs / Iwanami Series in Modern Mathematics
This is the English translation of the original Japanese book. In these books, Number Theory 1, 2, 3, we introduce core theories in modern number theory, such as class field theory, Iwasawa theory, the theory of modular forms, etc.
Vol.1: Fermat's Dream
In this volume, 'Fermat's Dream', core theories in modern number theory are introduced. Developments are given in elliptic curves, ?-adic numbers, the ?-function, and the number fields. This work presents an elegant perspective on the wonder of numbers.
Vol. 2: Introduction to Class Field Theory
This, the second of three related volumes on number theory, is the English translation of the original Japanese book. Here, the idea of class field theory, a highlight in algebraic number theory, is first described with many concrete examples. A detailed account of proofs is thoroughly exposited in the final chapter. The authors also explain the local-global method in number theory, including the use of ideles and adeles.
Vol. 3: Iwasawa Theory and Modular Forms
This is the third of three related volumes on number theory. 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 establishes far-reaching relations between a ?-adic analytic zeta function and a determinant defined from a Galois action on some ideal class groups. This book also contains a short exposition on the arithmetic of elliptic curves and the proof of Fermat's last theorem by Wiles. Together with the first two volumes, this book is a good resource for anyone learning or teaching modern algebraic number theory.
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