整数与数论探秘：GMT239初步

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"GMT239 数论1 是一本关于数论的研究生教材，该书深入探讨整数的性质，属于纯粹数学的一个分支。数论关注的是整数在方程（如丢番图方程）中的角色，以及通过解析函数（如黎曼ζ函数）来揭示质数和其他整数的特性。通过数论的研究，我们可以建立实数与有理数之间的联系，利用有理数对实数进行逼近，这就是丢番图逼近。" 数论作为数学的核心领域，其研究内容广泛且深奥。在标题提及的"GMT239 数论1"中，可能涵盖了以下几个关键知识点： 1. 丢番图方程：这是一种涉及整数解的代数方程，通常难以求解。数论家们致力于寻找这些方程的整数解，或者证明不存在这样的解，这对于理解整数结构有着重要意义。 2. 黎曼ζ函数：由伯恩哈德·黎曼提出的这个函数在数论中扮演着核心角色，它的零点分布与素数的分布紧密相关。通过分析黎曼ζ函数，可以推导出素数定理等重要结果。 3. 数的性质：数论研究整数的基本性质，如奇偶性、同余类、分解质因数等。这些问题看似简单，但实际上隐藏着丰富的数学结构。 4. 有理数与实数的关系：数论不仅关注整数，也关注如何用有理数来近似实数。丢番图逼近理论探讨了如何找到最接近某个实数的有理数，这在计算和几何学中有实际应用。 5. 解析数论：这是数论的一个分支，使用解析函数的方法来研究数论问题。例如，黎曼ζ函数就属于解析数论的范畴。 6. 代数数论：这个领域研究整数环上的代数结构，如理想、类群和伽罗瓦理论，它们与整数的性质密切相关。 7. 素数理论：素数是数论研究的核心，素数定理描述了素数在自然数中的分布规律。 8. 代数几何中的数论应用：数论的一些问题可以通过代数几何的方法来解决，例如，椭圆曲线在数论中的应用，与费马最后定理等著名问题有关。 9. 模形式：这些特殊类型的复解析函数在数论中有着深远的影响，它们与椭圆曲线、模群和自守形式等概念紧密相连。 10. 表示论与 Lie 群：这些理论在理解整数的性质上也有重要作用，特别是对于整数模的李代数的研究。 "GMT239 数论1"这本书可能包含以上多个方面的内容，为学习者提供了深入了解整数性质和相关理论的途径。通过阅读这本书，读者可以系统地学习数论的基础和高级概念，进一步探索数学的奥秘。

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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

− Dy

= N ........... 354

6.4 Diophantine Equations of Degree 3 . . . . . . . . . . . . . . . . . . . . . . . 357

6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

6.4.2 The Equation ax

+ by

+ cz

= 0: Local Solubility . . . 359

6.4.3 The Equation ax

+ by

+ cz

= 0: Number Fields . . . . 362

6.4.4 The Equation ax

+ by

+ cz

=0:

Hyperelliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

6.4.5 The Equation x

+ y

+ cz

=0.................... 373

6.4.6 Sums of Two or More Cubes . . . . . . . . . . . . . . . . . . . . . . . 376

6.4.7 Skolem’s Equations x

+ dy

=1 ................... 385

6.4.8 Special Cases of Skolem’s Equations . . . . . . . . . . . . . . . . 386

6.4.9 The Equations y

= x

± 1 in Rational Numbers . . . . . 387

6.5 The Equations ax

+ by

+ cz

=0andax

+ by

+ cz

= 0 . 389

6.5.1 The Equation ax

+ by

+ cz

= 0: Local Solubility . . . 389

6.5.2 The Equations x

± y

= z

and x

+2y

= z

....... 391

6.5.3 The Equation ax

+ by

+ cz

= 0: Elliptic Curves . . . . 392

6.5.4 The Equation ax

+ by

+ cz

= 0: Special Cases . . . . . 393

6.5.5 The Equation ax

+ by

+ cz

=0.................. 396

6.6 The Fermat Quartics x

+ y

= cz

....................... 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

= x

+ 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

+1=4x

and y

+ y +1=3x

. . . 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

+ y

= z

................... 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

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

+ y

=2z

. . . 480

7.3 Elliptic Curves over C, R, k(T ), F

,andK

............... 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

............................ 494

7.3.5 Constant Elliptic Curves over R[[T ]]: Formal Groups . . 500

7.3.6 Reduction of Elliptic Curves over K

............... 505

7.3.7 The p-adic Filtration for Elliptic Curves over K

..... 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

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

(1,x) for Small x ................ 575

8.5.5 Computation of Γ

(1,x)forLargex ................ 580

8.5.6 The Famous Curve y

+ y = x

− 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

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

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

(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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