费马大定理笔记(英文)

Quasi-historical introduction The cases n =2 and n =4.The Parisian Academy in the 1840s. Notes: Some details. Descent.Algebraic numbers and integers. Remarks on unique factorizationA digression. Notes:Continued fractions.Plagiarism. Elementary methods Sophie Germain, Abel's formulas, Mirimanoff-Wieferich,.. Notes: Fermat's Theorem.Bernoulli numbers. Euler-Maclaurin. Pseudoprimes. Fermat numbers. Mersenne primes. Cranks. Kummer's arguments Proof of the FLT for regular primes. Notes: Some remarks for undergraduates on elementary algebra. Equivalence relations. Why do we believe Wiles？ More quasi-historyRantings. Work on the FLT this century. Notes: Euler's conjecture. The growing of the "gap”. Diophantus and Fermat What the study of diophantine equations is really all about. Notes: The chord and tangent method.Examples. A child's introduction to elliptic functions For a precocious child. Notes: Discriminants. Local and global Some remarks on p-adic numbers. Notes: The Riemann C-function.Much more on p-adic numbers. Curves Particularly,about elliptic curves. Notes: Minimal model. Semisimplicity of the Frey curve. Birational equivalence. Modular forms Some formulas and assertions. Notes: More formulas.The discriminant function. The Modularity ConjectureAn attempt at an explanation. Notes: What's in a name？ The functional equation Poisson summation: 9-functions. Notes: Details. Hecke operators. Zeta functions and L-series Introduction to the Birch-Swinnerton-Dyer Conjectures. Notes: Hasse's Theorem. The ABC-Conjecture Darmon and Granville's Generalized Fermat Equation. Notes: Hawkins primes.The Generalized Fermat Conjecture. Heights Remarks on the Mordell-Weil Theorem. Notes: Lehmer's Question. Elliptic curves of high rank. Class number of imaginary quadratic number fieldsThe proof of Goldfeld-Gross-Zagier. Notes: Composition of quadratic forms. Tate-Shafarevitch group. Jacobian. Heegner points. Wiles'proof Not the commutative algebra, of course. Notes: Some details. Appendices Remarks on Fermat's Last Theorem For those who only want to pretend to have read the rest of this book. "The Devil and Simon Flagg", by Arthur PorgesThe devil fails where Wiles will succeed. “Math Riots Prove Fun Incalculable",by Eric ZornIs the FLT truly as important as sport？ Index 编辑手记