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 $\bbfQ$ 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. Indexed articles: {\it Stevens, Glenn}, An overview of the proof of Fermat’s Last Theorem, 1-16 [

Zbl 0916.11017] {\it Silverman, Joseph H.}, A survey of the arithmetic theory of elliptic curves, 17-40 [

Zbl 0906.11027] {\it Rohrlich, David E.}, Modular curves, Hecke correspondences, and $L$-functions, 41-100 [

Zbl 0897.11019] {\it Washington, Lawrence C.}, Galois cohomology, 101-120 [

Zbl 0928.12003] {\it Tate, John}, Finite flat group schemes, 121-154 [

Zbl 0924.14024] {\it Gelbart, Stephen}, Three lectures on the modularity of $\overline\rho_{E,3}$ and the Langlands reciprocity conjecture, 155-207 [

Zbl 0902.11016] {\it Edixhoven, Bas}, Serre’s conjecture, 209-242 [

Zbl 0918.11023] {\it Mazur, Barry}, An introduction to the deformation theory of Galois representations, 243-311 [

Zbl 0901.11015] {\it De Smit, Bart; Lenstra, Hendrik W.jun.}, Explicit construction of universal deformation rings, 313-326 [

Zbl 0907.13010] {\it Tilouine, Jacques}, Hecke algebras and the Gorenstein property, 327-342 [

Zbl 1155.11330] {\it De Smit, Bart; Rubin, Karl; Schoof, René}, Criteria for complete intersections, 343-355 [

Zbl 0903.13003] {\it Diamond, Fred; Ribet, Kenneth A.}, $\ell$-adic modular deformations and Wiles’s “Main Conjecture”, 357-371 [

Zbl 0919.11041] {\it Conrad, Brian}, The flat deformation functor, 373-420 [

Zbl 0927.11037] {\it de Shalit, Ehud}, Hecke rings and universal deformation rings, 421-445 [

Zbl 1044.11578] {\it Silverberg, Alice}, Explicit families of elliptic curves with prescribed $\text{mod } N$ representations, 447-461 [

Zbl 0912.11024] {\it Rubin, Karl}, Modularity of mod 5 representations, 463-474 [

Zbl 0914.11030] {\it Diamond, Fred}, An extension of Wiles’ results, 475-489 [

Zbl 0917.11021] {\it Diamond, Fred; Kramer, Kenneth}, Appendix: Classification of $\overline\rho_{E,\ell}$ by the $j$-invariant of $E$, 491-498 [

Zbl 0914.11029] {\it Lenstra, H.W.jun.; Stevenhagen, P.}, Class field theory and the first case of Fermat’s last theorem, 499-503 [

Zbl 0891.11012] {\it Rosen, Michael}, Remarks on the history of Fermat’s Last Theorem 1844 to 1984, 505-525 [

Zbl 0893.11011] {\it Frey, Gerhard}, On ternary equations of Fermat type and relations with elliptic curves, 527-548 [

Zbl 0976.11027] {\it Darmon, Henri}, Wiles’ theorem and the arithmetic of elliptic curves, 549-569 [

Zbl 0919.11043]