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This 2-volume set (Fermat's Last Theorem: Basic Tools and Fermat's
Last Theorem: The Proof) presents in full detail the proof of
Fermat's Last Theorem given by Wiles and Taylor. With these two
books, the reader will be able to see the whole picture of the
proof to appreciate one of the deepest achievements in the history
of mathematics. Crucial arguments, including the so-called 3-5
trick, R=T theorem, etc., are explained in depth. The proof relies
on basic background materials in number theory and arithmetic
geometry, such as elliptic curves, modular forms, Galois
representations, deformation rings, modular curves over the integer
rings, Galois cohomology, etc. The first four topics are crucial
for the proof of Fermat's Last Theorem; they are also very
important as tools in studying various other problems in modern
algebraic number theory. In order to facilitate understanding the
intricate proof, an outline of the whole argument is described in
the first preliminary chapter of the first volume.
This is the second volume of the book on the proof of Fermat's Last
Theorem by Wiles and Taylor (the first volume is published in the
same series; see MMONO/243). Here the detail of the proof announced
in the first volume is fully exposed. The book also includes basic
materials and constructions in number theory and arithmetic
geometry that are used in the proof. In the first volume the
modularity lifting theorem on Galois representations has been
reduced to properties of the deformation rings and the Hecke
modules. The Hecke modules and the Selmer groups used to study
deformation rings are constructed, and the required properties are
established to complete the proof. The reader can learn basics on
the integral models of modular curves and their reductions modulo
$p$ that lay the foundation of the construction of the Galois
representations associated with modular forms. More background
materials, including Galois cohomology, curves over integer rings,
the Neron models of their Jacobians, etc., are also explained in
the text and in the appendices.
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.
This book, together with the companion volume, Fermat's Last
Theorem: The Proof, presents in full detail the proof of Fermat's
Last Theorem given by Wiles and Taylor. With these two books, the
reader will be able to see the whole picture of the proof to
appreciate one of the deepest achievements in the history of
mathematics. Crucial arguments, including the so-called $3$-$5$
trick, $R=T$ theorem, etc., are explained in depth. The proof
relies on basic background materials in number theory and
arithmetic geometry, such as elliptic curves, modular forms, Galois
representations, deformation rings, modular curves over the integer
rings, Galois cohomology, etc. The first four topics are crucial
for the proof of Fermat's Last Theorem; they are also very
important as tools in studying various other problems in modern
algebraic number theory. The remaining topics will be treated in
the second book to be published in the same series in 2014. In
order to facilitate understanding the intricate proof, an outline
of the whole argument is described in the first preliminary
chapter, and more details are summarised in later chapters.
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 establishes far-reaching relations between a
$p$-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.
This book, 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. Basic properties of zeta and
$L$-functions are established and used to prove the prime number
theorem and the Dirichlet theorem on prime numbers in arithmetic
progressions. With this book, the reader can enjoy the beauty of
numbers and obtain fundamental knowledge of modern number theory.
The translation of the first volume was published as Number Theory
1: Fermat's Dream, Translations of Mathematical Monographs (Iwanami
Series in Modern Mathematics), vol. 186, American Mathematical
Society, 2000.
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