Number Theory
Number Theory
Number Theory
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xivTable of Contents6.11 First Results on Catalan’s Equation . . . . . . . . . . . . . . . . . . . . . . 4436.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4436.11.2 The Theorems of Nagell and Ko Chao . . . . . . . . . . . . . . 4456.11.3 Some Lemmas on Binomial Series . . . . . . . . . . . . . . . . . . 4466.11.4 Proof of Cassels’s Theorem 6.11.5 . . . . . . . . . . . . . . . . . . 4486.12 Congruent <strong>Number</strong>s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.12.1 Reduction to an Elliptic Curve . . . . . . . . . . . . . . . . . . . . . 4516.12.2 Use of the Birch and Swinnerton-Dyer Conjecture . . . . 4526.12.3 Tunnell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4546.13 Some Unsolved Diophantine Problems. . . . . . . . . . . . . . . . . . . . . 4556.14 Exercises for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4577. Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4657.1 Introduction and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4657.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4657.1.2 Weierstrass Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4657.1.3 Degenerate Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . 4677.1.4 The Group Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4707.1.5 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4727.2 Transformations into Weierstrass Form . . . . . . . . . . . . . . . . . . . . 4747.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 4747.2.2 Transformation of the Intersection of Two Quadrics. . . 4757.2.3 Transformation of a Hyperelliptic Quartic . . . . . . . . . . . 4767.2.4 Transformation of a General Nonsingular Cubic . . . . . . 4777.2.5 Example: the Diophantine Equation x 2 + y 4 = 2z 4 . . . . 4797.3 Elliptic Curves over C, R, k(T), F q , and K p . . . . . . . . . . . . . . . 4827.3.1 Elliptic Curves over C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4827.3.2 Elliptic Curves over R . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4847.3.3 Elliptic Curves over k(T) . . . . . . . . . . . . . . . . . . . . . . . . . . 4867.3.4 Elliptic Curves over F q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4947.3.5 Constant Elliptic Curves over R[[T]]: Formal Groups . . 4997.3.6 Reduction of Elliptic Curves over K p . . . . . . . . . . . . . . . 5047.3.7 The p-adic Filtration for Elliptic Curves over K p . . . . . 5067.4 Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5128. Diophantine Aspects of Elliptic Curves . . . . . . . . . . . . . . . . . . . 5178.1 Elliptic Curves over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5178.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5178.1.2 Basic Results and Conjectures . . . . . . . . . . . . . . . . . . . . . 5188.1.3 Computing the Torsion Subgroup . . . . . . . . . . . . . . . . . . 5248.1.4 Computing the Mordell–Weil Group . . . . . . . . . . . . . . . . 5288.1.5 The Naïve and Canonical Heights . . . . . . . . . . . . . . . . . . 5298.2 Description of 2-Descent with Rational 2-Torsion . . . . . . . . . . . 5328.2.1 The Fundamental 2-Isogeny. . . . . . . . . . . . . . . . . . . . . . . . 5328.2.2 Description of the Image of φ . . . . . . . . . . . . . . . . . . . . . . 534