简介
"The central theme of this graduate-level number theory textbook is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The first is the local aspect: one can do analysis in p-adic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second aspect is the global aspect: the use of number fields, and in particular of class groups and unit groups. The third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject, and embodies in a beautiful way the local and global aspects of Diophantine problems. In fact, these functions are defined through the local aspects of the problems, but their analytic behavior is intimately linked to the global aspects. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included 5 appendices on these techniques. These appendices were written by Henri Cohen, Yann Bugeaud, Maurice Mignotte, Sylvain Duquesne, and Samir Siksek, and contain material on the use of Galois representations, the superfermat equation, Mihailescu's proof of Catalan's Conjecture, and applications of linear forms in logarithms. The book contains 530 exercises of varying difficulty from immediate consequences of the main text to research problems, and contain many important additional results." -- Cover.
目录
Table Of Contents:
Preface
1. Introduction to Diophantine Equations 1
1.1 Introduction 1
1.1.1 Examples of Diophantine Problems 1
1.1.2 Local Methods 4
1.1.3 Dimensions 6
1.2 Exercises for Chapter 1 8
Part I. Tools
2. Abelian Groups, Lattices, and Finite Fields 11
2.1 Finitely Generated Abelian Groups 11
2.1.1 Basic Results 11
2.1.2 Description of Subgroups 16
2.1.3 Characters of Finite Abelian Groups 17
2.1.4 The Groups (Z/mZ)* 20
2.1.5 Dirichlet Characters 25
2.1.6 Gauss Sums 30
2.2 The Quadratic Reciprocity Law 33
2.2.1 The Basic Quadratic Reciprocity Law 33
2.2.2 Consequences of the Basic Quadratic Reciprocity Law 36
2.2.3 Gauss's Lemma and Quadratic Reciprocity 39
2.2.4 Real Primitive Characters 43
2.2.5 The Sign of the Quadratic Gauss Sum 45
2.3 Lattices and the Geometry of Numbers 50
2.3.1 Definitions 50
2.3.2 Hermite's Inequality 53
2.3.3 LLL-Reduced Bases 55
2.3.4 The LLL Algorithms 58
2.3.5 Approximation of Linear Forms 60
2.3.6 Minkowski's Convex Body Theorem 63
2.4 Basic Properties of Finite Fields 65
2.4.1 General Properties of Finite Fields 65
2.4.2 Galois Theory of Finite Fields 69
2.4.3 Polynomials over Finite Fields 71
2.5 Bounds for the Number of Solutions in Finite Fields 72
2.5.1 The Chevalley鈥揥arning Theorem 72
2.5.2 Gauss Sums for Finite Fields 73
2.5.3 Jacobi Sums for Finite Fields 79
2.5.4 The Jacobi Sums J(Chi;1,Χ2) 82
2.5.5 The Number of Solutions of Diagonal Equations 87
2.5.6 The Weil Bounds 90
2.5.7 The Weil Conjectures (Dengue's Theorem) 92
2.6 Exercises for Chapter 2 93
3. Basic Algebraic Number Theory 101
3.1 Field-Theoretic Algebraic Number Theory 101
3.1.1 Galois Theory 101
3.1.2 Number Fields 106
3.1.3 Examples 108
3.1.4 Characteristic Polynomial, Norm, Trace 109
3.1.5 Noether's Lemma 110
3.1.6 The Basic Theorem of Kummer Theory 111
3.1.7 Examples of the Use of Kummer Theory 114
3.1.8 Artin Schreier Theory 115
3.2 The Normal Basis Theorem 117
3.2.1 Linear Independence and Hilbert's Theorem 90 117
3.2.2 The Normal Basis Theorem in the Cyclic Case 119
3.2.3 Additive Polynomials 120
3.2.4 Algebraic Independence of Homomorphisms 121
3.2.5 The Normal Basis Theorem 123
3.3 Ring-Theoretic Algebraic Number Theory 124
3.3.1 Gauss's Lemma on Polynomials 124
3.3.2 Algebraic Integers 125
3.3.3 Ring of Integers and Discriminant 128
3.3.4 Ideals and Units 130
3.3.5 Decomposition of Primes and Ramification 132
3.3.6 Galois Properties of Prime Decomposition 134
3.4 Quadratic Fields 136
3.4.1 Field-Theoretic and Basic Ring-Theoretic Properties 136
3.4.2 Results and Conjectures on Class and Unit Groups 138
3.5 Cyclotomic Fields 140
3.5.1 Cyclotomic Polynomials 140
3.5.2 Field-Theoretic Properties of Q(ζn) 144
3.5.3 Ring-Theoretic Properties 146
3.5.4 The Totally Real Subfield of Q(ζpk) 148
3.6 Stickelberger's Theorem 150
3.6.1 Introduction and Algebraic Setting 150
3.6.2 Instantiation of Gauss Sums 151
3.6.3 Prime Ideal Decomposition of Gauss Sums 154
3.6.4 The Stickelberger Ideal 160
3.6.5 Diagonalization of the Stickelberger Element 163
3.6.6 The Eisenstein Reciprocity Law 165
3.7 The Hasse鈥揇avenport Relations 170
3.7.1 Distribution Formulas 171
3.7.2 The Hasse鈥揇avenport Relations 173
3.7.3 The Zeta Function of a Diagonal Hypersurface 177
3.8 Exercises for Chapter 3 179
4. p-adic Fields 183
4.1 Absolute Values and Completions 183
4.1.1 Absolute Values 183
4.1.2 Archimedean Absolute Values 184
4.1.3 Non-Archimedean and Ultrametric Absolute Values 188
4.1.4 Ostrowski's Theorem and the Product Formula 190
4.1.5 Completions 192
4.1.6 Completions of a Number Field 195
4.1.7 Hensel's Lemmas 199
4.2 Analytic Functions in p-adic Fields 205
4.2.1 Elementary Properties 205
4.2.2 Examples of Analytic Functions 208
4.2.3 Application of the Artin鈥揌asse Exponential 217
4.2.4 Mahler Expansions 220
4.3 Additive and Multiplicative Structures 224
4.3.1 Concrete Approach 224
4.3.2 Basic Reductions 225
4.3.3 Study of the Groups Ui 229
4.3.4 Study of the Group U1 231
4.3.5 The Group K*/Kp*虏 234
4.4 Extensions of p-adic Fields 235
4.4.1 Preliminaries on Local Field Norms 235
4.4.2 Krasner's Lemma 238
4.4.3 General Results on Extensions 239
4.4.4 Applications of the Cohomology of Cyclic Groups 242
4.4.5 Characterization of Unramified Extensions 249
4.4.6 Properties of Unramified Extensions 251
4.4.7 Totally Ramified Extensions 253
4.4.8 Analytic Representations of pth Roots of Unity 254
4.4.9 Factorizations in Number Fields 258
4.4.10 Existence of the Field Cp 260
4.4.11 Some Analysis in Cp 263
4.5 The Theorems of Strassmann and Weierstrass 266
4.5.1 Strassmann's Theorem 266
4.5.2 Krasner Analytic Functions 267
4.5.3 The Weierstrass Preparation Theorem 270
4.5.4 Applications of Strassmann's Theorem 272
4.6 Exercises for Chapter 4 275
5. Quadratic Forms and Local-Global Principles 285
5.1 Basic Results on Quadratic Forms 285
5.1.1 Basic Properties of Quadratic Modules 286
5.1.2 Contiguous Bases and Witt's Theorem 288
5.1.3 Translations into Results on Quadratic Forms 291
5.2 Quadratic Forms over Finite and Local Fields 294
5.2.1 Quadratic Forms over Finite Fields 294
5.2.2 Definition of the Local Hilbert Symbol 295
5.2.3 Main Properties of the Local Hilbert Symbol 296
5.2.4 Quadratic Forms over Qp 300
5.3 Quadratic Forms over Q 303
5.3.1 Global Properties of the Hilbert Symbol 303
5.3.2 Statement of the Hasse-Minkowski Theorem 305
5.3.3 The Hasse-Minkowski Theorem for n less than or equal to 2 306
5.3.4 The Hasse Minkowski Theorem for n = 3 307
5.3.5 The Hasse-Minkowski Theorem for n = 4 308
5.3.6 The Hasse Minkowski Theorem for N > or equal to 5 309
5.4 Consequences of the Hasse Minkowski Theorem 310
5.4.1 General Results 310
5.4.2 A Result of Davenport and Cassels 311
5.4.3 Universal Quadratic Forms 312
5.4.4 Sums of Squares 314
5.5 The Hasse Norm Principle 318
5.6 The Hasse Principle for Powers 321
5.6.1 A General Theorem on Powers 391
5.6.2 The Hasse Principle for Powers 324
5.7 Soule Counterexamples to the Hasse Principle 326
5.8 Exercises for Chapter 5 329
Part II. Diophantine Equations
6. Some Diophantine Equations 335
6.1 Introduction 335
6.1.1 The Use of Finite Fields 335
6.1.2 Local Methods 337
6.1.3 Global Methods 337
6.2 Diophantine Equations of Degree 1 339
6.3 Diophantine Equations of Degree 2 341
6.3.1 The General Homogeneous Equation 341
6.3.2 The Homogeneous Ternary Quadratic Equation 343
6.3.3 Computing a Particular Solution 347
6.3.4 Examples of Homogeneous Ternary Equations 352
6.3.5 The Pell-Fermat Equation x虏 鈥?Dy虏 = N 354
6.4 Diophantine Equations of Degree 3 357
6.4.1 Introduction 358
6.4.2 The Equation axP byP czP = 0: Local Solubility 359
6.4.3 The Equation axv +be + czp 0: Number Fields 362
6.4.4 The Equation axP + be + czP = 0: Hyperelliptic Curves 368
6.4.5 The Equation x鲁 + y鲁 + cz鲁 = 0 373
6.4.6 Sums of Two or More Cubes 376
6.4.7 Skolem's Equations x鲁 + dy鲁 = 1 385
6.4.8 Special Cases of Skolem's Equations 386
6.4.9 The Equations y虏 = x鲁 + 1 in Rational Numbers 387
6.5 The Equations ax4 + by4 + cz虏 = 0 and ax6 + by鲁 + cz虏 = 0 389
6.5.1 The Equation ax4 + by4 + cz虏 = 0: Local Solubility 389
6.5.2 The Equations x4 + y4 z2 and x4 + 2y4 = z虏 391
6.5.3 The Equation ax4 卤 by4 + cz虏 = 0: Elliptic Curves 392
6.5.4 The Equation ax4 卤 by4 + cz虏 = 0: Special Cases 393
6.5.5 The Equation ax6 + by鲁 + cz虏 = 0 396
6.6 The Fermat Quartics x4 + y4 = cz4 397
6.6.1 Local Solubility 398
6.6.2 Global Solubility: Factoring over Number Fields 400
6.6.3 Global Solubility: Coverings of Elliptic Curves 407
6.6.4 Conclusion, and a Small Table 409
6.7 The Equation y虏 = xn + t 410
6.7.1 General Results 411
6.7.2 The Case p = 3 414
6.7.3 The Case p = 5 416
6.7.4 Application of the Biln鈥擧anrot -Voutier Theorem 417
6.7.5 Special Cases with Fixed t 418
6.7.6 The Equations ty虏 + 1 = 4xP and y虏 + y + 1 = 3xp 420
6.8 Linear Recurring Sequences 421
6.8.1 Squares in the Fibonacci and Lucas Sequences 421
6.8.2 The Square Pyramid Problem 424
6.9 Fermat's "Last Theorem" xn + yn = zn 427
6.9.1 Introduction 427
6.9.2 General Prime n: The First Case 428
6.9.3 Congruence Criteria 429
6.9.4 The Criteria of Wendt and Germain 430
6.9.5 Kummer's Criterion: Regular Primes 431
6.9.6 The Criteria of Furtw盲ngler and Wieferich 434
6.9.7 General Prime n: The Second Case 435
6.10 An Example of Runge's Method 439
6.11 First Results on Catalan's Equation 442
6.11.1 Introduction 442
6.11.2 The Theorems of Nagel' and Ko Chao 444
6.11.3 Some Lemmas on Binomial Series 446
6.11.4 Proof of Cassels's Theorem 6.11.5 447
6.12 Congruent Numbers 450
6.12.1 Reduction to an Elliptic Curve 451
6.12.2 The Use of the Birch and Swinnerton-Dyer Conjecture 452
6.12.3 Tunnell's Theorem 453
6.13 Some Unsolved Diophantine Problems 455
6.14 Exercises for Chapter 6 456
7. Elliptic Curves 465
7.1 Introduction and Definitions 465
7.1.1 Introduction 465
7.1.2 Weierstrass Equations 465
7.1.3 Degenerate Elliptic Curves 467
7.1.4 The Group Law 470
7.1.5 Isogenies 472
7.2 Transformations into Weierstrass Form 474
7.2.1 Statement of the Problem 474
7.2.2 Transformation of the Intersection of Two Quadrics 475
7.2.3 Transformation of a Hyperelliptic Quartic 476
7.2.4 'Transformation of a General Nonsingular Cubic 477
7.2.5 Example: The Diophantine Equation x虏 + y4 = 2z4 480
7.3 Elliptic Curves over C, R, k(T), Fq, and Kp 482
7.3.1 Elliptic Curves over C 482
7.3.2 Elliptic Curves over R 484
7.3.3 Elliptic Curves over k(T) 486
7.3.4 Elliptic Curves over Fq 494
7.3.5 Constant Elliptic Curves over R[[T]]: Formal Groups 500
7.3.6 Reduction of Elliptic Curves over Kp 505
7.3.7 The p-adic Filtration for Elliptic Curves over Kp 507
7.4 Exercises for Chapter 7 512
8. Diophantine Aspects of Elliptic Curves 517
8.1 Elliptic Curves over Q 517
8.1.1 Introduction 517
8.1.2 Basic Results and Conjectures 518
8.1.3 Computing the Torsion Subgroup 524
8.1.4 Computing the Mordell鈥揥eil Group 528
8.1.5 The Na茂ve and Canonical Heights 529
8.2 Description of 2-Descent with Rational 2-Torsion 532
8.2.1 The Fundamental 2-Isogeny 532
8.2.2 Description of the Image of 酶 534
8.2.3 The Fundamental 2-Descent Map 535
8.2.4 Practical Use of 2-Descent with 2-Isogenies 538
8.2.5 Examples of 2-Descent using 2-Isogenies 542
8.2.6 An Example of Second Descent 546
8.3 Description of General 2-Descent 548
8.3.1 The Fundamental 2-Descent Map 548
8.3.2 The T-Selmer Group of a Number Field 550
8.3.3 Description of the Image of a 酶 552
8.3.4 Practical Use of 2-Descent in the General Case 554
8.3.5 Examples of General 2-Descent 555
8.4 Description of 3-Descent with Rational 3-Torsion Subgroup 557
8.4.1 Rational 3-Torsion Subgroups 557
8.4.2 The Fundamental 3-Isogeny 558
8.4.3 Description of the Image of 酶 560
8.4.4 The Fundamental 3-Descent Map 563
8.5 The Use of L(E, s) 564
8.5.1 Introduction 564
8.5.2 The Case of Complex Multiplication 565
8.5.3 Numerical Computation of L(r)(E, 1) 572
8.5.4 Computation of Γ(1, x) for Small x 575
8.5.5 Computation of Γ(1, x) for Large x 580
8.5.6 The Famous Curve y虏 + y = X鲁 - 7x + 6 582
8.6 The Heegner Point Method 584
8.6.1 Introduction and the Modular Parametrization 584
8.6.2 Heegner Points and Complex Multiplication 586
8.6.3 The Use of the Theorem of Gross Zagier 589
8.6.4 Practical Use of the Heegner Point Method 591
8.6.5 Improvements to the Basic Algorithm, in Brief 596
8.6.6 A Complete Example 598
8.7 Computation of Integral Points 600
8.7.1 Introduction 600
8.7.2 An Upper Bound for the Elliptic Logarithm on E(Z) 601
8.7.3 Lower Bounds for Linear Forms in Elliptic Logarithms 603
8.7.4 A Complete Example 605
8.8 Exercises for Chapter 8 607
Bibliography 615
Index of Notation 625
Index of Names 633
General Index 639
Preface
1. Introduction to Diophantine Equations 1
1.1 Introduction 1
1.1.1 Examples of Diophantine Problems 1
1.1.2 Local Methods 4
1.1.3 Dimensions 6
1.2 Exercises for Chapter 1 8
Part I. Tools
2. Abelian Groups, Lattices, and Finite Fields 11
2.1 Finitely Generated Abelian Groups 11
2.1.1 Basic Results 11
2.1.2 Description of Subgroups 16
2.1.3 Characters of Finite Abelian Groups 17
2.1.4 The Groups (Z/mZ)* 20
2.1.5 Dirichlet Characters 25
2.1.6 Gauss Sums 30
2.2 The Quadratic Reciprocity Law 33
2.2.1 The Basic Quadratic Reciprocity Law 33
2.2.2 Consequences of the Basic Quadratic Reciprocity Law 36
2.2.3 Gauss's Lemma and Quadratic Reciprocity 39
2.2.4 Real Primitive Characters 43
2.2.5 The Sign of the Quadratic Gauss Sum 45
2.3 Lattices and the Geometry of Numbers 50
2.3.1 Definitions 50
2.3.2 Hermite's Inequality 53
2.3.3 LLL-Reduced Bases 55
2.3.4 The LLL Algorithms 58
2.3.5 Approximation of Linear Forms 60
2.3.6 Minkowski's Convex Body Theorem 63
2.4 Basic Properties of Finite Fields 65
2.4.1 General Properties of Finite Fields 65
2.4.2 Galois Theory of Finite Fields 69
2.4.3 Polynomials over Finite Fields 71
2.5 Bounds for the Number of Solutions in Finite Fields 72
2.5.1 The Chevalley鈥揥arning Theorem 72
2.5.2 Gauss Sums for Finite Fields 73
2.5.3 Jacobi Sums for Finite Fields 79
2.5.4 The Jacobi Sums J(Chi;1,Χ2) 82
2.5.5 The Number of Solutions of Diagonal Equations 87
2.5.6 The Weil Bounds 90
2.5.7 The Weil Conjectures (Dengue's Theorem) 92
2.6 Exercises for Chapter 2 93
3. Basic Algebraic Number Theory 101
3.1 Field-Theoretic Algebraic Number Theory 101
3.1.1 Galois Theory 101
3.1.2 Number Fields 106
3.1.3 Examples 108
3.1.4 Characteristic Polynomial, Norm, Trace 109
3.1.5 Noether's Lemma 110
3.1.6 The Basic Theorem of Kummer Theory 111
3.1.7 Examples of the Use of Kummer Theory 114
3.1.8 Artin Schreier Theory 115
3.2 The Normal Basis Theorem 117
3.2.1 Linear Independence and Hilbert's Theorem 90 117
3.2.2 The Normal Basis Theorem in the Cyclic Case 119
3.2.3 Additive Polynomials 120
3.2.4 Algebraic Independence of Homomorphisms 121
3.2.5 The Normal Basis Theorem 123
3.3 Ring-Theoretic Algebraic Number Theory 124
3.3.1 Gauss's Lemma on Polynomials 124
3.3.2 Algebraic Integers 125
3.3.3 Ring of Integers and Discriminant 128
3.3.4 Ideals and Units 130
3.3.5 Decomposition of Primes and Ramification 132
3.3.6 Galois Properties of Prime Decomposition 134
3.4 Quadratic Fields 136
3.4.1 Field-Theoretic and Basic Ring-Theoretic Properties 136
3.4.2 Results and Conjectures on Class and Unit Groups 138
3.5 Cyclotomic Fields 140
3.5.1 Cyclotomic Polynomials 140
3.5.2 Field-Theoretic Properties of Q(ζn) 144
3.5.3 Ring-Theoretic Properties 146
3.5.4 The Totally Real Subfield of Q(ζpk) 148
3.6 Stickelberger's Theorem 150
3.6.1 Introduction and Algebraic Setting 150
3.6.2 Instantiation of Gauss Sums 151
3.6.3 Prime Ideal Decomposition of Gauss Sums 154
3.6.4 The Stickelberger Ideal 160
3.6.5 Diagonalization of the Stickelberger Element 163
3.6.6 The Eisenstein Reciprocity Law 165
3.7 The Hasse鈥揇avenport Relations 170
3.7.1 Distribution Formulas 171
3.7.2 The Hasse鈥揇avenport Relations 173
3.7.3 The Zeta Function of a Diagonal Hypersurface 177
3.8 Exercises for Chapter 3 179
4. p-adic Fields 183
4.1 Absolute Values and Completions 183
4.1.1 Absolute Values 183
4.1.2 Archimedean Absolute Values 184
4.1.3 Non-Archimedean and Ultrametric Absolute Values 188
4.1.4 Ostrowski's Theorem and the Product Formula 190
4.1.5 Completions 192
4.1.6 Completions of a Number Field 195
4.1.7 Hensel's Lemmas 199
4.2 Analytic Functions in p-adic Fields 205
4.2.1 Elementary Properties 205
4.2.2 Examples of Analytic Functions 208
4.2.3 Application of the Artin鈥揌asse Exponential 217
4.2.4 Mahler Expansions 220
4.3 Additive and Multiplicative Structures 224
4.3.1 Concrete Approach 224
4.3.2 Basic Reductions 225
4.3.3 Study of the Groups Ui 229
4.3.4 Study of the Group U1 231
4.3.5 The Group K*/Kp*虏 234
4.4 Extensions of p-adic Fields 235
4.4.1 Preliminaries on Local Field Norms 235
4.4.2 Krasner's Lemma 238
4.4.3 General Results on Extensions 239
4.4.4 Applications of the Cohomology of Cyclic Groups 242
4.4.5 Characterization of Unramified Extensions 249
4.4.6 Properties of Unramified Extensions 251
4.4.7 Totally Ramified Extensions 253
4.4.8 Analytic Representations of pth Roots of Unity 254
4.4.9 Factorizations in Number Fields 258
4.4.10 Existence of the Field Cp 260
4.4.11 Some Analysis in Cp 263
4.5 The Theorems of Strassmann and Weierstrass 266
4.5.1 Strassmann's Theorem 266
4.5.2 Krasner Analytic Functions 267
4.5.3 The Weierstrass Preparation Theorem 270
4.5.4 Applications of Strassmann's Theorem 272
4.6 Exercises for Chapter 4 275
5. Quadratic Forms and Local-Global Principles 285
5.1 Basic Results on Quadratic Forms 285
5.1.1 Basic Properties of Quadratic Modules 286
5.1.2 Contiguous Bases and Witt's Theorem 288
5.1.3 Translations into Results on Quadratic Forms 291
5.2 Quadratic Forms over Finite and Local Fields 294
5.2.1 Quadratic Forms over Finite Fields 294
5.2.2 Definition of the Local Hilbert Symbol 295
5.2.3 Main Properties of the Local Hilbert Symbol 296
5.2.4 Quadratic Forms over Qp 300
5.3 Quadratic Forms over Q 303
5.3.1 Global Properties of the Hilbert Symbol 303
5.3.2 Statement of the Hasse-Minkowski Theorem 305
5.3.3 The Hasse-Minkowski Theorem for n less than or equal to 2 306
5.3.4 The Hasse Minkowski Theorem for n = 3 307
5.3.5 The Hasse-Minkowski Theorem for n = 4 308
5.3.6 The Hasse Minkowski Theorem for N > or equal to 5 309
5.4 Consequences of the Hasse Minkowski Theorem 310
5.4.1 General Results 310
5.4.2 A Result of Davenport and Cassels 311
5.4.3 Universal Quadratic Forms 312
5.4.4 Sums of Squares 314
5.5 The Hasse Norm Principle 318
5.6 The Hasse Principle for Powers 321
5.6.1 A General Theorem on Powers 391
5.6.2 The Hasse Principle for Powers 324
5.7 Soule Counterexamples to the Hasse Principle 326
5.8 Exercises for Chapter 5 329
Part II. Diophantine Equations
6. Some Diophantine Equations 335
6.1 Introduction 335
6.1.1 The Use of Finite Fields 335
6.1.2 Local Methods 337
6.1.3 Global Methods 337
6.2 Diophantine Equations of Degree 1 339
6.3 Diophantine Equations of Degree 2 341
6.3.1 The General Homogeneous Equation 341
6.3.2 The Homogeneous Ternary Quadratic Equation 343
6.3.3 Computing a Particular Solution 347
6.3.4 Examples of Homogeneous Ternary Equations 352
6.3.5 The Pell-Fermat Equation x虏 鈥?Dy虏 = N 354
6.4 Diophantine Equations of Degree 3 357
6.4.1 Introduction 358
6.4.2 The Equation axP byP czP = 0: Local Solubility 359
6.4.3 The Equation axv +be + czp 0: Number Fields 362
6.4.4 The Equation axP + be + czP = 0: Hyperelliptic Curves 368
6.4.5 The Equation x鲁 + y鲁 + cz鲁 = 0 373
6.4.6 Sums of Two or More Cubes 376
6.4.7 Skolem's Equations x鲁 + dy鲁 = 1 385
6.4.8 Special Cases of Skolem's Equations 386
6.4.9 The Equations y虏 = x鲁 + 1 in Rational Numbers 387
6.5 The Equations ax4 + by4 + cz虏 = 0 and ax6 + by鲁 + cz虏 = 0 389
6.5.1 The Equation ax4 + by4 + cz虏 = 0: Local Solubility 389
6.5.2 The Equations x4 + y4 z2 and x4 + 2y4 = z虏 391
6.5.3 The Equation ax4 卤 by4 + cz虏 = 0: Elliptic Curves 392
6.5.4 The Equation ax4 卤 by4 + cz虏 = 0: Special Cases 393
6.5.5 The Equation ax6 + by鲁 + cz虏 = 0 396
6.6 The Fermat Quartics x4 + y4 = cz4 397
6.6.1 Local Solubility 398
6.6.2 Global Solubility: Factoring over Number Fields 400
6.6.3 Global Solubility: Coverings of Elliptic Curves 407
6.6.4 Conclusion, and a Small Table 409
6.7 The Equation y虏 = xn + t 410
6.7.1 General Results 411
6.7.2 The Case p = 3 414
6.7.3 The Case p = 5 416
6.7.4 Application of the Biln鈥擧anrot -Voutier Theorem 417
6.7.5 Special Cases with Fixed t 418
6.7.6 The Equations ty虏 + 1 = 4xP and y虏 + y + 1 = 3xp 420
6.8 Linear Recurring Sequences 421
6.8.1 Squares in the Fibonacci and Lucas Sequences 421
6.8.2 The Square Pyramid Problem 424
6.9 Fermat's "Last Theorem" xn + yn = zn 427
6.9.1 Introduction 427
6.9.2 General Prime n: The First Case 428
6.9.3 Congruence Criteria 429
6.9.4 The Criteria of Wendt and Germain 430
6.9.5 Kummer's Criterion: Regular Primes 431
6.9.6 The Criteria of Furtw盲ngler and Wieferich 434
6.9.7 General Prime n: The Second Case 435
6.10 An Example of Runge's Method 439
6.11 First Results on Catalan's Equation 442
6.11.1 Introduction 442
6.11.2 The Theorems of Nagel' and Ko Chao 444
6.11.3 Some Lemmas on Binomial Series 446
6.11.4 Proof of Cassels's Theorem 6.11.5 447
6.12 Congruent Numbers 450
6.12.1 Reduction to an Elliptic Curve 451
6.12.2 The Use of the Birch and Swinnerton-Dyer Conjecture 452
6.12.3 Tunnell's Theorem 453
6.13 Some Unsolved Diophantine Problems 455
6.14 Exercises for Chapter 6 456
7. Elliptic Curves 465
7.1 Introduction and Definitions 465
7.1.1 Introduction 465
7.1.2 Weierstrass Equations 465
7.1.3 Degenerate Elliptic Curves 467
7.1.4 The Group Law 470
7.1.5 Isogenies 472
7.2 Transformations into Weierstrass Form 474
7.2.1 Statement of the Problem 474
7.2.2 Transformation of the Intersection of Two Quadrics 475
7.2.3 Transformation of a Hyperelliptic Quartic 476
7.2.4 'Transformation of a General Nonsingular Cubic 477
7.2.5 Example: The Diophantine Equation x虏 + y4 = 2z4 480
7.3 Elliptic Curves over C, R, k(T), Fq, and Kp 482
7.3.1 Elliptic Curves over C 482
7.3.2 Elliptic Curves over R 484
7.3.3 Elliptic Curves over k(T) 486
7.3.4 Elliptic Curves over Fq 494
7.3.5 Constant Elliptic Curves over R[[T]]: Formal Groups 500
7.3.6 Reduction of Elliptic Curves over Kp 505
7.3.7 The p-adic Filtration for Elliptic Curves over Kp 507
7.4 Exercises for Chapter 7 512
8. Diophantine Aspects of Elliptic Curves 517
8.1 Elliptic Curves over Q 517
8.1.1 Introduction 517
8.1.2 Basic Results and Conjectures 518
8.1.3 Computing the Torsion Subgroup 524
8.1.4 Computing the Mordell鈥揥eil Group 528
8.1.5 The Na茂ve and Canonical Heights 529
8.2 Description of 2-Descent with Rational 2-Torsion 532
8.2.1 The Fundamental 2-Isogeny 532
8.2.2 Description of the Image of 酶 534
8.2.3 The Fundamental 2-Descent Map 535
8.2.4 Practical Use of 2-Descent with 2-Isogenies 538
8.2.5 Examples of 2-Descent using 2-Isogenies 542
8.2.6 An Example of Second Descent 546
8.3 Description of General 2-Descent 548
8.3.1 The Fundamental 2-Descent Map 548
8.3.2 The T-Selmer Group of a Number Field 550
8.3.3 Description of the Image of a 酶 552
8.3.4 Practical Use of 2-Descent in the General Case 554
8.3.5 Examples of General 2-Descent 555
8.4 Description of 3-Descent with Rational 3-Torsion Subgroup 557
8.4.1 Rational 3-Torsion Subgroups 557
8.4.2 The Fundamental 3-Isogeny 558
8.4.3 Description of the Image of 酶 560
8.4.4 The Fundamental 3-Descent Map 563
8.5 The Use of L(E, s) 564
8.5.1 Introduction 564
8.5.2 The Case of Complex Multiplication 565
8.5.3 Numerical Computation of L(r)(E, 1) 572
8.5.4 Computation of Γ(1, x) for Small x 575
8.5.5 Computation of Γ(1, x) for Large x 580
8.5.6 The Famous Curve y虏 + y = X鲁 - 7x + 6 582
8.6 The Heegner Point Method 584
8.6.1 Introduction and the Modular Parametrization 584
8.6.2 Heegner Points and Complex Multiplication 586
8.6.3 The Use of the Theorem of Gross Zagier 589
8.6.4 Practical Use of the Heegner Point Method 591
8.6.5 Improvements to the Basic Algorithm, in Brief 596
8.6.6 A Complete Example 598
8.7 Computation of Integral Points 600
8.7.1 Introduction 600
8.7.2 An Upper Bound for the Elliptic Logarithm on E(Z) 601
8.7.3 Lower Bounds for Linear Forms in Elliptic Logarithms 603
8.7.4 A Complete Example 605
8.8 Exercises for Chapter 8 607
Bibliography 615
Index of Notation 625
Index of Names 633
General Index 639
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