Notes on Fermat's Last Theorem

Read also the publisher's blurb.

Alf van der Poorten, Notes on Fermat's Last Theorem
Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley--Interscience, January, 1996
222 + xvi pages
ISBN 0-471-06261-8
Library of Congress Call Number QA 244.V36 1996

Everything you've been wanting to ask about number theory and Fermat's Last Theorem 
       but were afraid to admit you didn't know.

"Notes on Fermat's Last Theorem" 
was awarded the Association of American Publishers 
1996 Professional/Scholarly Publishing Award for Excellence in Mathematics.

Note that a second corrected and mildly revised edition is in preparation.

"The poetry far excels that normally found in math books". H W Lenstra

"I love the book. Thanks for writing it. If you're ever in the Cotswolds come and stay". K B MD

"Hype and false promotion". "Lack of scholarship". Serge Lang

"... it should be bedtime reading for every mathematician". Ram Murty

"It is well-worth the two stars that the Monthly mini-reviews gave it ('A=B' only got one star)". Doron Zeilberger

"... polished, eccentric, opinionated and inspiring ... " Andrew Granville [see Amer. Math. Monthly (2), (1999)]

"There's lots of good mathematics books .... but not many that are fun!" Armand Borel

Table of Contents

Introduction
Biographical Remarks


Lecture I. Quasi-historical introduction
The cases $n=2$ and $n=4$. The Parisian Academy in the 1840 s.
Notes: Some details. Descent. Algebraic numbers and integers.
1--10

Lecture II. Remarks on unique factorization
A digression.
Notes: Continued fractions. Plagiarism
11--18

Lecture III. Elementary methods
Sophie Germain, Abel's formulas, Mirimanoff--Wieferich, ...
Notes: Fermat's Theorem. Bernoulli numbers. Euler--Maclaurin.
Pseudoprimes. Fermat numbers. Mersenne primes. Cranks.
19--30

Lecture IV. Kummer's arguments
Proof of the FLT for regular primes.
Notes: Some remarks for undergraduates on elementary algebra. 
Equivalence relations.
31--39

Lecture V. Why do we believe Wiles? More quasi-history
 
Rantings. Work on the FLT this century.
Notes: Euler's conjecture. The growing of the ``gap''.  
41--50

Lecture VI. Diophantus and Fermat
What the study of diophantine equations is really all about. 
Notes: The chord and tangent method. Examples.
51--63

Lecture VII. A child's introduction to elliptic functions
 
For a precocious child.
Notes: Discriminants.
65--73

Lecture VIII. Local and global
Some remarks on $p$-adic numbers.
Notes: The Riemann $\zeta$-function. Much more on $p$-adic numbers.
75--88

Lecture IX. Curves
Particularly, about elliptic curves.
Notes: Minimal model. Semisimplicity of the Frey curve. 
Birational equivalence.
89--101

Lecture X. Modular forms
Some formulas and assertions.
Notes: More formulas. The discriminant function.
103--111

Lecture XI. The Modularity Conjecture
An attempt at an explanation.
Notes: What's in a name?
113--122

Lecture XII. The functional equation
Poisson summation; $\vartheta$--functions.
Notes: Details. Hecke operators.
123--133

Lecture XIII. Zeta functions and $L$--series
Introduction to the Birch--Swinnerton-Dyer Conjectures.
Notes: Hasse's Theorem.
135--141

Lecture XIV. The ABC--Conjecture
Darmon and Granville's Generalized Fermat Equation.
Notes: Hawkins primes. The Generalized Fermat Conjecture.
143--150

Lecture XV. Heights
Remarks on the Mordell--Weil Theorem.
Notes: Lehmer's Question. Elliptic curves of high rank.
151--159

Lecture XVI. Class number of imaginary quadratic number fields
The proof of Goldfeld--Gross--Zagier.
Notes: Composition of quadratic forms. 
Tate--Shafarevitch group. Jacobian. Heegner points.
161--176

Lecture XVII. Wiles' proof
Not the commutative algebra, of course.
Notes: Some details.
177--188

Appendices

Appendix A. Remarks on Fermat's Last Theorem
For those who only want to pretend to have looked at the rest of this book.
187--199

Appendix B. "The Devil and Simon Flagg", by Arthur Porges
The devil fails where Wiles will succeed.
201--206

Appendix C. "Math Riots Prove Fun Incalculable", by Eric Zorn
Is the FLT truly as important as sport?
207--209

Index
211--222




 












Alf van der Poorten 
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Killara  Australia 2071
Fax: +61 2 9850 8114
 
Mobile: +61 4 1826 3129

   alf@math.mq.edu.au 
  






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