zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Modular forms and Fermat’s last theorem. Papers from a conference, Boston, MA, USA, August 9–18, 1995. (English) Zbl 0878.11004
New York, NY: Springer. xviii, 582 p. DM 89.00; öS 649.70; sFr. 81.00; £34.00; $ 49.95 (1997).

The articles of this volume will be reviewed individually. Introduction: The chapters of this book are expanded versions of the lectures given at the BU conference. They are intended to introduce the many ideas and techniques used by Wiles in his proof that every (semi-stable) elliptic curve over is modular, and to explain how Wiles’ result combined with Ribet’s theorem implies the validity of Fermat’s Last Theorem.

The first chapter contains an overview of the complete proof, and it is followed by introductory chapters surveying the basic theory of elliptic curves (Chapter II), modular functions and curves (Chapter III), Galois cohomology (Chapter IV), and finite group schemes (Chapter V). Next we turn to the representation theory which lies at the core of Wiles’ proof. Chapter VI gives an introduction to automorphic representations and the Langlands-Tunnel theorem, which provides the crucial first step that a certain mod 3 representation is modular. Chapter VII describes Serre’s conjectures and the known cases which give the link between modularity of elliptic curves and Fermat’s Last Theorem. After this come chapters on deformations of Galois representations (Chapter VIII) and universal deformation rings (Chapter IX), followed by chapters on Hecke algebras (Chapter X) and complete intersections (Chapter XI). Chapters XII and XIV contain the heart of Wiles’ proof, with a brief interlude (Chapter XIII) devoted to representability of the flat deformation functor. The final step in Wiles’ proof, the so-called “3-5 shift,” is discussed in Chapters XV and XVI, and Diamond’s relaxation of the semi-stability condition is described in Chapter XVII. The volume concludes by looking both backward and forward in time, with two chapters (Chapter XVIII and XIX) describing some of the “pre-modular” history of Fermat’s Last Theorem, and two chapters (Chapters XX and XXI) placing Wiles’ theorem into a more general Diophantine context and giving some ideas of possible future applications.

As the preceding brief summary will have made clear, the proof of Wiles’ theorem is extremely intricate and draws on tools from many areas of mathematics. The editors hope that this volume will help everyone, student and professional mathematician alike, who wants to study the details of what is surely one of the most memorable mathematical achievements of this century.


MSC:
11-06Proceedings of conferences (number theory)
00B25Proceedings of conferences of miscellaneous specific interest
11D41Higher degree diophantine equations
11G18Arithmetic aspects of modular and Shimura varieties
11F11Holomorphic modular forms of integral weight
14HxxAlgebraic curves