This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.
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"Das Buch ist eine hervorragende Basis, um in die moderne Arithmetik einzudringen, und zeigt zugleich in besonderer Eindringlichkeit die Schlagkraft der in den letzten Jahrzehnten entwickelten abstrakten Methoden, um konkrete diophantische Probleme wie die Fermatsche Vermutung zu lösen." DMV Jahresbericht, Bd. 103, Heft 1, 2001
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Kenneth A. Ribet studied mathematics at Brown University and Harvard University. He received his PhD in 1973 from Harvard, where his advisor was John Tate. After three years of teaching in Princeton and two years of research in Paris, Ribet joined the University of California, Berkeley faculty in 1978.
Ribet is known for his work in number theory and algebraic geometry. He played a prominent role in the proof of Fermat's Last Theorem by showing that this statement was a logical consequence of a conjecture about elliptic curves. (Andrew Wiles proved this conjecture in 1995, thereby obtaining Fermat's Last Theorem as a corollary.)
Ribet was elected to the American Academy of Arts and Sciences in 1997 and the National Academy of Sciences in 2000. He was awarded the Fermat Prize in 1989 and received an honorary PhD from Brown University in 1998. Ribet was inducted as a Vigneron d'honneur by the Jurade de Saint Emilion in 1988.
Ribet is known for his work in number theory and algebraic geometry. He played a prominent role in the proof of Fermat's Last Theorem by showing that this statement was a logical consequence of a conjecture about elliptic curves. (Andrew Wiles proved this conjecture in 1995, thereby obtaining Fermat's Last Theorem as a corollary.)
Ribet was elected to the American Academy of Arts and Sciences in 1997 and the National Academy of Sciences in 2000. He was awarded the Fermat Prize in 1989 and received an honorary PhD from Brown University in 1998. Ribet was inducted as a Vigneron d'honneur by the Jurade de Saint Emilion in 1988.
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19 of 27 people found the following review helpful:
5.0 out of 5 stars
Yet another application of elliptic curves...,
By Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews (VINE VOICE) (HALL OF FAME REVIEWER) (REAL NAME)
This review is from: Modular Forms and Fermat's Last Theorem (Hardcover)
The successful proof of Fermat's Last Theorem by Andrew Wiles was probably the most widely publicized mathematical result of the 20th century. And once again, among their numerous other applications, elliptic curves are employed in the proof. The book is a compilation of articles written by first-class mathematicians, the reading of which will give one a thorough understanding of the proof, along with an overview of some very interesting topics in number theory and algebraic geometry. The reader who undertakes an understanding of the proof already no doubt has a substantial amount of 'mathematical maturity', and no review, no matter how complete, would influence greatly such a reader. Suffice it to say then that this book is excellent, and even a reader interested solely in elliptic curves and modular forms could benefit greatly from the reading of this book.
3 of 44 people found the following review helpful:
3.0 out of 5 stars
Highly recommended,
By Jens Leopold (Tiefensee, Germany) - See all my reviews
This review is from: Modular Forms and Fermat's Last Theorem (Hardcover)
This item is very instructively, not only for "real" mathematicians. Of course, sometimes it's very difficult to "read". It gives me pleasure to own the proof of FLT.
3 of 63 people found the following review helpful:
5.0 out of 5 stars
Great?!?!,
By A Customer
This review is from: Modular Forms and Fermat's Last Theorem (Hardcover)
This book might be good if you like number theory. But if you're an analyst who hates number theory or a brick-layer, then this book is probably not meant for you. I hope you found this review helpful. Have a nice day.
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Inside This Book
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Key Phrases - Statistically Improbable Phrases (SIPs):
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main conjecture, flat deformation functor, analytic group isomorphism, normal subgroup scheme, order one satisfying, finite flat group schemes, reciprocity conjecture, unramified quotient, fiber representation, automorphic cuspidal representation, universal deformation ring, base change lift, height conjecture, closed subgroup scheme, supersingular reduction, generalized elliptic curves, supersingular points, multiplicative reduction, trivial centralizer, irrationality degree, modular deformations, potential good reduction, complete local noetherian ring, strict equivalence class, bottom horizontal arrow Key Phrases - Capitalized Phrases (CAPs): (learn more)
Fermat's Last Theorem, Annals of Math, Lecture Notes, New York, International Press, Pure Math, Duke Math, Raynaud F-module, Springer Verlag, Global Galois, Princeton University Press, Azumaya Algebra, Academic Press, Schur's Lemma, Local Fields, Carayol's Lemma, Langlands Program, Modular Functions of One Variable, Hong Kong, Séminaire Bourbaki, Gerhart Frey, Proof of Property, Graduate Texts, Cambridge Univ, London Math Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
main conjecture, flat deformation functor, analytic group isomorphism, normal subgroup scheme, order one satisfying, finite flat group schemes, reciprocity conjecture, unramified quotient, fiber representation, automorphic cuspidal representation, universal deformation ring, base change lift, height conjecture, closed subgroup scheme, supersingular reduction, generalized elliptic curves, supersingular points, multiplicative reduction, trivial centralizer, irrationality degree, modular deformations, potential good reduction, complete local noetherian ring, strict equivalence class, bottom horizontal arrow Key Phrases - Capitalized Phrases (CAPs): (learn more)
Fermat's Last Theorem, Annals of Math, Lecture Notes, New York, International Press, Pure Math, Duke Math, Raynaud F-module, Springer Verlag, Global Galois, Princeton University Press, Azumaya Algebra, Academic Press, Schur's Lemma, Local Fields, Carayol's Lemma, Langlands Program, Modular Functions of One Variable, Hong Kong, Séminaire Bourbaki, Gerhart Frey, Proof of Property, Graduate Texts, Cambridge Univ, London Math Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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