Volume I
Preface ......................................................... v
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 ( /m)* ............................. 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-Warning 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(χ1, 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 (Deligne'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  (ζn) ............ 144
3.5.3. Ring-Theoretic Properties ...................... 146
3.5.4. The Totally Real Subfield of (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 Hassc Davenport Relations ......................... 170
3.7.1. Distribution Formulas .......................... 171
3.7.2. The Hasse-Davenport 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 Hasse 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 Kp*/Kp*2 .............................. 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  p ..................... 260
4.4.11. Some Analysis in p ........................... 263
4.5. The Theorems of Strassmann and Weierstrass ............ 266
4.5.1. Strassmarm'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 P ........................ 300
5.3. Quadratic Forms over ................................ 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 ≤ 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 ≥ 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 .................... 321
5.6.2. The Hasse Principle for Powers ................. 324
5.7. Some 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 x2 - Dy2 = 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 axp + byp + czp = 0:
Number Fields .................................. 362
6.4.4. The Equation axp + byp + czp = 0:
Hyperelliptic Curves ........................... 368
6.4.5. The Equation x3 + y3 + cz3 = 0 .................. 373
6.4.6. Sums of Two or More Cubes ...................... 376
6.4.7. Skolem's Equations x3 + dy3 = 1 ................ 385
6.4.8. Special Cases of Skolem's Equations ............ 386
6.4.9. The Equations y2 = x3 ± 1 in Rational
Numbers ........................................ 387
6.5. The Equations ax4 + by4 + cz2 = 0 and
ax6 + by3 + cz2 =0 .................................... 389
6.5.1. The Equation ax4 + by4 + cz2 = 0:
Local Solubility ............................... 389
6.5.2. The Equations x4 ± y4 = z2 and x4 + 2y4 = z2 .... 391
6.5.3. The Equation ax4 + by4 + cz2 = 0:
Elliptic Curves ................................ 392
6.5.4. The Equation ax4 + by4 + cz2 = 0:
Special Cases .................................. 393
6.5.5. The Equation ax6 + by3 + cz2 = 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 y2 = 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 Bilu-Hanrot-Voutier
Theorem ........................................ 417
6.7.5. Special Cases with Fixed t ..................... 418
6.7.6. The Equations ty2 + 1 = 4xp and
у2 + у + 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. Knmmer's Criterion: Regular Primes ............. 431
6.9.1. The Criteria of Furtwangler 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 Nagell and Ко 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 Swirmerton-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
x2 + y4 = 2z4 ................................... 480
7.3. Elliptic Curves over ,  , k(T),  q and Kp ............ 482
7.3.1. Elliptic Curves over ......................... 482
7.3.2. Elliptic Curves over ......................... 484
7.3.3. Elliptic Curves over k(T) ...................... 486
7.3.4. Elliptic Curves over q ........................ 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 ................................ 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-Weil Group ............... 528
8.1.5. The Naive 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  .................. 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 α .................. 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 .................. 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 Гr(1, х) for Small x ............ 575
8.5.5. Computation of Гr(1, x) for Large x ............ 580
8.5.6. The Famous Curve у2 + у = x3 - 7x + 6 .......... 582
8.6. The Heegner Point Method .............................. 584
8.6.1. Introduction and the Modular
Parameterization ............................... 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() ......................................... 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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