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首页探索数学魅力:数字理论经典入门
《数论:第一卷》是一本深入浅出的数学专著,它属于Graduate Texts in Mathematics系列,旨在向读者介绍并探索数字理论领域的核心概念和理论。本书将带领读者进入数学世界的一个迷人分支,使他们领略到数学魅力的深度和复杂性。 数论是数学的一个基础且历史悠久的领域,主要研究整数及其相互关系。该书中可能涵盖的内容包括基本的数论概念,如素数、同余关系、欧几里得算法、迪利克雷定理、费马大定理等,这些都是数论中的基石。此外,它也可能探讨更高级的主题,如椭圆曲线、模形式、算术几何以及现代密码学中的应用,如RSA加密算法背后的数学原理。 作为研究生教材,这本书可能采用递进式的方式,从基础的算术原理出发,逐步引入更复杂的理论工具和技术,如模环和模算术,以便读者在掌握基础知识的同时,逐步构建坚实的数论理论体系。书中可能会包含大量的例题和习题,帮助读者巩固理解和实践所学。 在编纂者阵容中,包括了诸如S.Axler、K.A.Ribet等知名数学家,他们的贡献表明了该书的权威性和专业性。《数论:第一卷》不仅适合正在攻读数学专业的研究生,对于对数学有深厚兴趣的业余爱好者,甚至是从事密码学或信息安全等领域工作的专业人士,都是一本不可或缺的学习资料。 通过阅读这本书,读者不仅能深入了解数论的基本概念,还能培养抽象思维能力、逻辑推理能力和问题解决技巧,这些都是数学研究和实际应用中至关重要的素质。《数论:第一卷》是任何希望在数论领域深化理解或进行独立研究的读者的理想选择。
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Table of Contents xv
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
2
− Dy
2
= N ........... 354
6.4 Diophantine Equations of Degree 3 . . . . . . . . . . . . . . . . . . . . . . . 357
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
6.4.2 The Equation ax
p
+ by
p
+ cz
p
= 0: Local Solubility . . . 359
6.4.3 The Equation ax
p
+ by
p
+ cz
p
= 0: Number Fields . . . . 362
6.4.4 The Equation ax
p
+ by
p
+ cz
p
=0:
Hyperelliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
6.4.5 The Equation x
3
+ y
3
+ cz
3
=0.................... 373
6.4.6 Sums of Two or More Cubes . . . . . . . . . . . . . . . . . . . . . . . 376
6.4.7 Skolem’s Equations x
3
+ dy
3
=1 ................... 385
6.4.8 Special Cases of Skolem’s Equations . . . . . . . . . . . . . . . . 386
6.4.9 The Equations y
2
= x
3
± 1 in Rational Numbers . . . . . 387
6.5 The Equations ax
4
+ by
4
+ cz
2
=0andax
6
+ by
3
+ cz
2
= 0 . 389
6.5.1 The Equation ax
4
+ by
4
+ cz
2
= 0: Local Solubility . . . 389
6.5.2 The Equations x
4
± y
4
= z
2
and x
4
+2y
4
= z
2
....... 391
6.5.3 The Equation ax
4
+ by
4
+ cz
2
= 0: Elliptic Curves . . . . 392
6.5.4 The Equation ax
4
+ by
4
+ cz
2
= 0: Special Cases . . . . . 393
6.5.5 The Equation ax
6
+ by
3
+ cz
2
=0.................. 396
6.6 The Fermat Quartics x
4
+ y
4
= cz
4
....................... 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
2
= x
n
+ 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 ty
2
+1=4x
p
and y
2
+ y +1=3x
p
. . . 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” x
n
+ y
n
= z
n
................... 427
6.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
6.9.2 General Prime n: The FirstCase................... 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
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xvi Table of Contents
6.9.6 The Criteria of Furtw¨anglerandWieferich........... 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 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
2
+ y
4
=2z
4
. . . 480
7.3 Elliptic Curves over C, R, k(T ), F
q
,andK
p
............... 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 F
q
............................ 494
7.3.5 Constant Elliptic Curves over R[[T ]]: Formal Groups . . 500
7.3.6 Reduction of Elliptic Curves over K
p
............... 505
7.3.7 The p-adic Filtration for Elliptic Curves over K
p
..... 507
7.4 ExercisesforChapter7 ................................. 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–Weil Group . . . . . . . . . . . . . . . . 528
8.1.5 The Na¨ıve and Canonical Heights . . . . . . . . . . . . . . . . . . 529
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Table of Contents xvii
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 φ ...................... 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 -SelmerGroupof aNumberField............. 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 φ ...................... 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
r
(1,x) for Small x ................ 575
8.5.5 Computation of Γ
r
(1,x)forLargex ................ 580
8.5.6 The Famous Curve y
2
+ y = x
3
− 7x +6 ............ 582
8.6 TheHeegnerPoint 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 ExercisesforChapter8 ................................. 607
Bibliography .............................................. 615
Index of Notation ........................................ 625
Index of Names ........................................... 633
General Index ............................................ 639
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xviii Table of Contents
Volume I I
Preface ................................................... v
Part III. Analytic Tools
9. Bernoulli Polynomials and the Gamma Function .......... 3
9.1 Bernoulli Numbers and Polynomials . . . . . . . . . . . . . . . . . . . . . . 3
9.1.1 Generating Functions for Bernoulli Polynomials . . . . . . 3
9.1.2 Further Recurrences for Bernoulli Polynomials . . . . . . . 10
9.1.3 Computing a Single Bernoulli Number . . . . . . . . . . . . . . 14
9.1.4 Bernoulli Polynomials and Fourier Series . . . . . . . . . . . . 16
9.2 Analytic Applications of Bernoulli Polynomials . . . . . . . . . . . . . 19
9.2.1 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9.2.2 The Euler–MacLaurin Summation Formula . . . . . . . . . . 21
9.2.3 The Remainder Term and the Constant Term . . . . . . . . 25
9.2.4 Euler–MacLaurin and the Laplace Transform . . . . . . . . 27
9.2.5 Basic Applications of the Euler–MacLaurin Formula . . 31
9.3 Applications to Numerical Integration . . . . . . . . . . . . . . . . . . . . . 35
9.3.1 Standard Euler–MacLaurin Numerical Integration . . . . 36
9.3.2 The Basic Tanh-Sinh Numerical Integration Method . . 37
9.3.3 General Doubly Exponential Numerical Integration . . . 39
9.4 χ-Bernoulli Numbers, Polynomials, and Functions . . . . . . . . . . 43
9.4.1 χ-Bernoulli Numbers and Polynomials . . . . . . . . . . . . . . 43
9.4.2 χ-Bernoulli Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9.4.3 The χ-Euler–MacLaurin Summation Formula . . . . . . . . 50
9.5 Arithmetic Properties of Bernoulli Numbers . . . . . . . . . . . . . . . 52
9.5.1 χ-PowerSums ................................... 52
9.5.2 The Generalized Clausen–von Staudt Congruence . . . . 61
9.5.3 The Voronoi Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.5.4 The Kummer Congruences . . . . . . . . . . . . . . . . . . . . . . . . 67
9.5.5 The Almkvist–Meurman Theorem . . . . . . . . . . . . . . . . . . 70
9.6 The Real and Complex Gamma Functions . . . . . . . . . . . . . . . . . 71
9.6.1 The Hurwitz Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . 71
9.6.2 Definition of the Gamma Function . . . . . . . . . . . . . . . . . . 77
9.6.3 Preliminary Results for the Study of Γ(s)............ 81
9.6.4 Properties of the Gamma Function . . . . . . . . . . . . . . . . . 84
9.6.5 Specific Properties of the Function ψ(s) ............. 95
9.6.6 Fourier Expansions of ζ(s, x) and log(Γ(x)).......... 100
9.7 IntegralTransforms..................................... 103
9.7.1 Generalities on Integral Transforms . . . . . . . . . . . . . . . . . 104
9.7.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.7.3 The Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
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Table of Contents xix
9.7.4 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.8 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.8.2 Integral Representations and Applications . . . . . . . . . . . 113
9.9 ExercisesforChapter9 ................................. 118
10. Dirichlet Series and L-Functions .......................... 151
10.1 Arithmetic Functions and Dirichlet Series. . . . . . . . . . . . . . . . . . 151
10.1.1 Operations on Arithmetic Functions . . . . . . . . . . . . . . . . 152
10.1.2 Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.1.3 Some Classical Arithmetical Functions . . . . . . . . . . . . . . 155
10.1.4 Numerical Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . 160
10.2 The Analytic Theory of L-Series . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.2.1 Simple Approaches to Analytic Continuation. . . . . . . . . 163
10.2.2 The Use of the Hurwitz Zeta Function ζ(s, x) ........ 168
10.2.3 The Functional Equation for the Theta Function . . . . . 169
10.2.4 The Functional Equation for Dirichlet L-Functions . . . 172
10.2.5 Generalized Poisson Summation Formulas . . . . . . . . . . . 177
10.2.6 Voronoi’s Error Term in the Circle Problem. . . . . . . . . . 182
10.3 Special Values of Dirichlet L-Functions . . . . . . . . . . . . . . . . . . . . 186
10.3.1 Basic Results on Special Values . . . . . . . . . . . . . . . . . . . . 186
10.3.2 Special Values of L-Functions and Modular Forms . . . . 193
10.3.3 The P´olya–Vinogradov Inequality . . . . . . . . . . . . . . . . . . 198
10.3.4 Bounds and Averages for L(χ, 1) ................... 200
10.3.5 Expansions of ζ(s) Around s = k ∈ Z
1
............. 205
10.3.6 Numerical Computation of Euler Products and Sums . 208
10.4 Epstein Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.4.1 The Nonholomorphic Eisenstein Series G(τ,s)........ 211
10.4.2 The Kronecker Limit Formula . . . . . . . . . . . . . . . . . . . . . . 213
10.5 Dirichlet Series Linked to Number Fields . . . . . . . . . . . . . . . . . . 216
10.5.1 The Dedekind Zeta Function ζ
K
(s) ................. 216
10.5.2 The Dedekind Zeta Function of Quadratic Fields . . . . . 219
10.5.3 Applications of the Kronecker Limit Formula . . . . . . . . 223
10.5.4 The Dedekind Zeta Function of Cyclotomic Fields . . . . 230
10.5.5 The Nonvanishing of L(χ, 1) ....................... 235
10.5.6 Application to Primes in Arithmetic Progression . . . . . 237
10.5.7 Conjectures on Dirichlet L-Functions . . . . . . . . . . . . . . . 238
10.6 Science Fiction on L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.6.1 Local L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.6.2 Global L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.7 The Prime Number Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.7.1 Estimates for ζ(s) ................................ 246
10.7.2 Newman’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
10.7.3 Iwaniec’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
10.8 Exercises for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
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