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Number theory: 2 Analytic and modern tools

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1. Verfasser:	Cohen, Henri 1947- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York, NY Springer 2007
Schriftenreihe:	Graduate texts in mathematics 240
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XXIII, 596 Seiten
ISBN:	9780387498942 9780387498935

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Datensatz im Suchindex

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adam_text	xviii Table of Contents Volume II 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 -Bemoulli Numbers, Polynomials, and Functions .......... 43 9.4.1 ^-Bernoulli Numbers and Polynomials .............. 43 9.4.2 ^-Bernoulli Functions ............................. 46 9.4.3 The x-Euler-MacLaurin Summation Formula ........ 50 9.5 Arithmetic Properties of Bernoulli Numbers ............... 52 9.5.1 х -Power Sums ................................... 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 F(s) ............ 81 9.6.4 Properties of the Gamma Function ................. 84 9.6.5 Specific Properties of the Function -ф(ѕ) ............. 95 9.6.6 Fourier Expansions of C(s,x) and log(r(x)) .......... 100 9.7 Integral Transforms ..................................... 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 Exercises for Chapter 9 .................................118 10. Dirichlet Series and ¿-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(3,χ) ........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 Pólya-Vinogradov Inequality ..................198 10.3.4 Bounds and Averages for L( , 1)...................200 10.3.5 Expansions of C(s) Around s — к Є Z^i .............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(t, 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 ÇK(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{ , 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 C(s) ................................246 10.7.2 Newman s Proof .................................250 10.7.3 Iwaniec s Proof ...................................254 10.8 Exercises for Chapter 10 ................................258 xx Table of Contents 11. p-adic Gamma and L-Functions ...........................275 11.1 Generalities on p-adic Functions ..........................275 11.1.1 Methods for Constructing p-adic Functions ..........275 11.1.2 A Brief Study of Volkenborn Integrals ...............276 11.2 The p-adic Hurwitz Zeta Functions .......................280 11.2.1 Teichmüller Extensions and Characters on Zp ........280 11.2.2 The p-adic Hurwitz Zeta Function for χ Є CZP .......281 11.2.3 The Function Cp(s,x) Around s - 1.................288 11.2.4 The p-adic Hurwitz Zeta Function for χ Є Zp ........290 11.3 p-adic L-Functions ......................................300 11.3.1 Dirichlet Characters in the p-adic Context ...........300 11.3.2 Definition and Basic Properties of p-adic L-Functions . 301 11.3.3 p-adic L-Functions at Positive Integers ..............305 11.3.4 χ -Power Sums Involving p-adic Logarithms ..........310 11.3.5 The Function Lp(x, s) Around s = 1................317 11.4 Applications of p-adic L-Functions ........................319 11.4.1 Integrality and Parity of L-Function Values ..........319 11.4.2 Bernoulli Numbers and Regular Primes .............324 11.4.3 Strengthening of the Almkvist-Meurman Theorem .. . 326 11.5 p-adic Log Gamma Functions ............................329 11.5.1 Diamond s p-adic Log Gamma Function .............330 11.5.2 Morita s p-adic Log Gamma Function ...............336 11.5.3 Computation of some p-adic Logarithms .............346 11.5.4 Computation of Limits of some Logarithmic Sums .... 356 11.5.5 Explicit Formulas for -фр(г/т) and -фр(х,г/т) .......359 11.5.6 Application to the Value of ¿ρ(χ, 1) ................361 11.6 Morita s p-adic Gamma Function .........................364 11.6.1 Introduction ..................................... 364 11.6.2 Definitions and Basic Results ...................... 365 11.6.3 Main Properties of the p-adic Gamma Function ...... 369 11.6.4 Mahler-Dwork Expansions Linked to Tp(x) .......... 375 11.6.5 Power Series Expansions Linked to Tp(x) ............ 378 11.6.6 The Jacobstahl-Kazandzidis Congruence ............ 380 11.7 The Gross-Koblitz Formula and Applications .............. 383 11.7.1 Statement and Proof of the Gross-Koblitz Formula ... 383 11.7.2 Application to L p(x, 0)............................389 11.7.3 Application to the Stickelberger Congruence .........390 11.7.4 Application to the Hasse-Davenport Product Relation 392 11.8 Exercises for Chapter 11 ................................395 Table of Contents Part IV. Modem Tools 12. Applications of Linear Forms in Logarithms ..............411 12.1 Introduction ...........................................411 12.1.1 Lower Bounds ...................................411 12.1.2 Applications to Diophantine Equations and Problems .413 12.1.3 A List of Applications ............................414 12.2 A Lower Bound for \|2m - 3n\| ............................415 12.3 Lower Bounds for the Trace of a™ ........................418 12.4 Pure Powers in Binary Recurrent Sequences ...............420 12.5 Greatest Prime Factors of Terms of Some Recurrent Se¬ quences ...............................................421 12.6 Greatest Prime Factors of Values of Integer Polynomials .....422 12.7 The Diophantine Equation axn - byn = с ..................423 12.8 Simultaneous Pell Equations .............................424 12.8.1 General Strategy .................................424 12.8.2 An Example in Detail .............................425 12.8.3 A General Algorithm .............................426 12.9 Catalan s Equation .....................................428 12.10 Thue Equations .......................................430 12.10.1 The Main Theorem ..............................430 12.10.2 Algorithmic Aspects .............................432 12.11 Other Classical Diophantine Equations ...................436 12.12 A Few Words on the Non- Archimedean Case ..............439 13. Rational Points on Higher-Genus Curves .................441 13.1 Introduction ...........................................441 13.2 The Jacobian ..........................................442 13.2.1 Functions on Curves ..............................443 13.2.2 Divisors .........................................444 13.2.3 Rational Divisors .................................445 13.2.4 The Group Law: Cantor s Algorithm ................446 13.2.5 The Group Law: The Geometric Point of View .......448 13.3 Rational Points on Hyperelliptic Curves ...................449 13.3.1 The Method of Dem yanenko-Manin ................449 13.3.2 The Method of Chabauty-Coieman .................452 13.3.3 Explicit Chabauty According to Flynn ..............453 13.3.4 When Chabauty Fails .............................455 13.3.5 Elliptic Curve Chabauty ..........................456 13.3.6 A Complete Example .............................459 xxii Table of Contents 14. The Super-Fermat Equation ..............................463 14.1 Preliminary Reductions .................................463 14.2 The Dihedral Cases (2,2, r) ..............................465 14.2.1 The Equation x2 - y2 = zr ........................ 465 14.2.2 The Equation x2 + y2 = zr ........................ 466 14.2.3 The Equations x2 + 3y2 = z3 and x2 + 3y2 = 4z3 ..... 466 14.3 The Tetrahedral Case (2,3,3)............................ 467 14.3.1 The Equation x3 + y3 = z2 ........................467 14.3.2 The Equation x3 + y3 = 2z2 .......................470 14.3.3 The Equation x3-2y3 = z2 .......................472 14.4 The Octahedral Case (2,3,4) ............................473 14.4.1 The Equation x2 - y4 = z3 ........................473 14.4.2 The Equation x2 + y4 = z3 ........................475 14.5 Invariants, Covariante, and Dessins d Enfants.............. 477 14.5.1 Dessins d Enfants, Klein Forms, and Covariants ......478 14.5.2 The Icosahedral Case (2,3,5)......................479 14.6 The Parabolic and Hyperbolic Cases ......................481 14.6.1 The Parabolic Case ............................... 481 14.6.2 General Results in the Hyperbolic Case ............. 482 14.6.3 The Equations x4 ± y4 = z3 ....................... 484 14.6.4 The Equation x4 + y4 = z5 ........................ 485 14.6.5 The Equation x6 - y4 = z2 ........................ 486 14.6.6 The Equation x4 - y6 = z2 ........................ 487 14.6.7 The Equation x6 + y4 = z2 ........................ 488 14.6.8 Further Results .................................. 489 14.7 Applications of Mason s Theorem ......................... 490 14.7.1 Mason s Theorem ................................491 14.7.2 Applications .....................................492 14.8 Exercises for Chapter 14 ................................493 15. The Modular Approach to Diophantine Equations ........495 15.1 Newforms .............................................495 15.1.1 Introduction and Necessary Software Tools ..........495 15.1.2 Newforms .......................................496 15.1.3 Rational Newforms and Elliptic Curves .............497 15.2 Ribet s Level-Lowering Theorem ..........................498 15.2.1 Definition of Arises From ........................498 15.2.2 Ribet s Level-Lowering Theorem ...................499 15.2.3 Absence of Isogenies..............................501 15.2.4 How to use Ribet s Theorem .......................503 15.3 Fermaťs Last Theorem and Similar Equations .............503 15.3.1 A Generalization of FLT ..........................504 15.3.2 E Arises from a Curve with Complex Multiplication . . 505 15.3.3 End of the Proof of Theorem 15.3.1.................506 15.3.4 The Equation x2 = yP + 2rzp for ρ ^ 7 and r > 2.....507 Table of Contents xxiii 15.3.5 The Equation χ2 = yP + zp for ρ ^ 7................509 15.4 An Occasional Bound for the Exponent ...................509 15.5 An Example of Serre-Mazur-Kraus .......................511 15.6 The Method of Kraus...................................514 15.7 Predicting Exponents of Constants .....................517 15.7.1 The Diophantine Equation x2 - 2 = yp ..............517 15.7.2 Application to the SMK Equation ..................521 15.8 Recipes for Some Ternary Diophantine Equations ...........522 15.8.1 Recipes for Signature (ρ,ρ,ρ) ......................523 15.8.2 Recipes for Signature (p,p,2) ......................524 15.8.3 Recipes for Signature (p, p, 3) ......................526 16. Catalan s Equation .......................................529 16.1 Mihăilescu s First Two Theorems .........................529 16.1.1 The First Theorem: Double Wieferich Pairs ..........530 16.1.2 The Equation (χΡ - )/{x - 1) = pyl ...............532 16.1.3 Mihăilescu s Second Theorem: ρ \| h~ and q h~ ......536 16.2 The + and - Subspaces and the Group S .................537 16.2.1 The + and - Subspaces ...........................538 16.2.2 The Group S ....................................540 16.3 Mihăilescu s Third Theorem: p < Aq2 and q < 4p2 ..........542 16.4 Mihăilescu s Fourth Theorem: ρ ξ 1 (mod q) or q = 1 (mod p) 547 16.4.1 Preliminaries on Commutative Algebra ..............547 16.4.2 Preliminaries on the Plus Part .....................549 16.4.3 Cyclotomic Units and Thaine s Theorem ............552 16.4.4 Preliminaries on Power Series ......................554 16.4.5 Proof of Mihăilescu s Fourth Theorem ..............557 16.4.6 Conclusion: Proof of Catalan s Conjecture ...........560 Bibliography ..............................................561 Index of Notation ........................................571 Index of Names ...........................................579 General Index ............................................585
adam_txt	xviii Table of Contents Volume II 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 -Bemoulli Numbers, Polynomials, and Functions . 43 9.4.1 ^-Bernoulli Numbers and Polynomials . 43 9.4.2 ^-Bernoulli Functions . 46 9.4.3 The x-Euler-MacLaurin Summation Formula . 50 9.5 Arithmetic Properties of Bernoulli Numbers . 52 9.5.1 х -Power Sums . 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 F(s) . 81 9.6.4 Properties of the Gamma Function . 84 9.6.5 Specific Properties of the Function -ф(ѕ) . 95 9.6.6 Fourier Expansions of C(s,x) and log(r(x)) . 100 9.7 Integral Transforms . 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 Exercises for Chapter 9 .118 10. Dirichlet Series and ¿-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(3,χ) .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 Pólya-Vinogradov Inequality .198 10.3.4 Bounds and Averages for L(\, 1).200 10.3.5 Expansions of C(s) Around s — к Є Z^i .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(t, 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 ÇK(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{\, 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 C(s) .246 10.7.2 Newman's Proof .250 10.7.3 Iwaniec's Proof .254 10.8 Exercises for Chapter 10 .258 xx Table of Contents 11. p-adic Gamma and L-Functions .275 11.1 Generalities on p-adic Functions .275 11.1.1 Methods for Constructing p-adic Functions .275 11.1.2 A Brief Study of Volkenborn Integrals .276 11.2 The p-adic Hurwitz Zeta Functions .280 11.2.1 Teichmüller Extensions and Characters on Zp .280 11.2.2 The p-adic Hurwitz Zeta Function for χ Є CZP .281 11.2.3 The Function Cp(s,x) Around s - 1.288 11.2.4 The p-adic Hurwitz Zeta Function for χ Є Zp .290 11.3 p-adic L-Functions .300 11.3.1 Dirichlet Characters in the p-adic Context .300 11.3.2 Definition and Basic Properties of p-adic L-Functions . 301 11.3.3 p-adic L-Functions at Positive Integers .305 11.3.4 χ -Power Sums Involving p-adic Logarithms .310 11.3.5 The Function Lp(x, s) Around s = 1.317 11.4 Applications of p-adic L-Functions .319 11.4.1 Integrality and Parity of L-Function Values .319 11.4.2 Bernoulli Numbers and Regular Primes .324 11.4.3 Strengthening of the Almkvist-Meurman Theorem . . 326 11.5 p-adic Log Gamma Functions .329 11.5.1 Diamond's p-adic Log Gamma Function .330 11.5.2 Morita's p-adic Log Gamma Function .336 11.5.3 Computation of some p-adic Logarithms .346 11.5.4 Computation of Limits of some Logarithmic Sums . 356 11.5.5 Explicit Formulas for -фр(г/т) and -фр(х,г/т) .359 11.5.6 Application to the Value of ¿ρ(χ, 1) .361 11.6 Morita's p-adic Gamma Function .364 11.6.1 Introduction . 364 11.6.2 Definitions and Basic Results . 365 11.6.3 Main Properties of the p-adic Gamma Function . 369 11.6.4 Mahler-Dwork Expansions Linked to Tp(x) . 375 11.6.5 Power Series Expansions Linked to Tp(x) . 378 11.6.6 The Jacobstahl-Kazandzidis Congruence . 380 11.7 The Gross-Koblitz Formula and Applications . 383 11.7.1 Statement and Proof of the Gross-Koblitz Formula . 383 11.7.2 Application to L'p(x, 0).389 11.7.3 Application to the Stickelberger Congruence .390 11.7.4 Application to the Hasse-Davenport Product Relation 392 11.8 Exercises for Chapter 11 .395 Table of Contents Part IV. Modem Tools 12. Applications of Linear Forms in Logarithms .411 12.1 Introduction .411 12.1.1 Lower Bounds .411 12.1.2 Applications to Diophantine Equations and Problems .413 12.1.3 A List of Applications .414 12.2 A Lower Bound for \|2m - 3n\| .415 12.3 Lower Bounds for the Trace of a™ .418 12.4 Pure Powers in Binary Recurrent Sequences .420 12.5 Greatest Prime Factors of Terms of Some Recurrent Se¬ quences .421 12.6 Greatest Prime Factors of Values of Integer Polynomials .422 12.7 The Diophantine Equation axn - byn = с .423 12.8 Simultaneous Pell Equations .424 12.8.1 General Strategy .424 12.8.2 An Example in Detail .425 12.8.3 A General Algorithm .426 12.9 Catalan's Equation .428 12.10 Thue Equations .430 12.10.1 The Main Theorem .430 12.10.2 Algorithmic Aspects .432 12.11 Other Classical Diophantine Equations .436 12.12 A Few Words on the Non- Archimedean Case .439 13. Rational Points on Higher-Genus Curves .441 13.1 Introduction .441 13.2 The Jacobian .442 13.2.1 Functions on Curves .443 13.2.2 Divisors .444 13.2.3 Rational Divisors .445 13.2.4 The Group Law: Cantor's Algorithm .446 13.2.5 The Group Law: The Geometric Point of View .448 13.3 Rational Points on Hyperelliptic Curves .449 13.3.1 The Method of Dem'yanenko-Manin .449 13.3.2 The Method of Chabauty-Coieman .452 13.3.3 Explicit Chabauty According to Flynn .453 13.3.4 When Chabauty Fails .455 13.3.5 Elliptic Curve Chabauty .456 13.3.6 A Complete Example .459 xxii Table of Contents 14. The Super-Fermat Equation .463 14.1 Preliminary Reductions .463 14.2 The Dihedral Cases (2,2, r) .465 14.2.1 The Equation x2 - y2 = zr . 465 14.2.2 The Equation x2 + y2 = zr . 466 14.2.3 The Equations x2 + 3y2 = z3 and x2 + 3y2 = 4z3 . 466 14.3 The Tetrahedral Case (2,3,3). 467 14.3.1 The Equation x3 + y3 = z2 .467 14.3.2 The Equation x3 + y3 = 2z2 .470 14.3.3 The Equation x3-2y3 = z2 .472 14.4 The Octahedral Case (2,3,4) .473 14.4.1 The Equation x2 - y4 = z3 .473 14.4.2 The Equation x2 + y4 = z3 .475 14.5 Invariants, Covariante, and Dessins d'Enfants. 477 14.5.1 Dessins d'Enfants, Klein Forms, and Covariants .478 14.5.2 The Icosahedral Case (2,3,5).479 14.6 The Parabolic and Hyperbolic Cases .481 14.6.1 The Parabolic Case . 481 14.6.2 General Results in the Hyperbolic Case . 482 14.6.3 The Equations x4 ± y4 = z3 . 484 14.6.4 The Equation x4 + y4 = z5 . 485 14.6.5 The Equation x6 - y4 = z2 . 486 14.6.6 The Equation x4 - y6 = z2 . 487 14.6.7 The Equation x6 + y4 = z2 . 488 14.6.8 Further Results . 489 14.7 Applications of Mason's Theorem . 490 14.7.1 Mason's Theorem .491 14.7.2 Applications .492 14.8 Exercises for Chapter 14 .493 15. The Modular Approach to Diophantine Equations .495 15.1 Newforms .495 15.1.1 Introduction and Necessary Software Tools .495 15.1.2 Newforms .496 15.1.3 Rational Newforms and Elliptic Curves .497 15.2 Ribet's Level-Lowering Theorem .498 15.2.1 Definition of "Arises From" .498 15.2.2 Ribet's Level-Lowering Theorem .499 15.2.3 Absence of Isogenies.501 15.2.4 How to use Ribet's Theorem .503 15.3 Fermaťs Last Theorem and Similar Equations .503 15.3.1 A Generalization of FLT .504 15.3.2 E Arises from a Curve with Complex Multiplication . . 505 15.3.3 End of the Proof of Theorem 15.3.1.506 15.3.4 The Equation x2 = yP + 2rzp for ρ ^ 7 and r > 2.507 Table of Contents xxiii 15.3.5 The Equation χ2 = yP + zp for ρ ^ 7.509 15.4 An Occasional Bound for the Exponent .509 15.5 An Example of Serre-Mazur-Kraus .511 15.6 The Method of Kraus.514 15.7 "Predicting Exponents of Constants" .517 15.7.1 The Diophantine Equation x2 - 2 = yp .517 15.7.2 Application to the SMK Equation .521 15.8 Recipes for Some Ternary Diophantine Equations .522 15.8.1 Recipes for Signature (ρ,ρ,ρ) .523 15.8.2 Recipes for Signature (p,p,2) .524 15.8.3 Recipes for Signature (p, p, 3) .526 16. Catalan's Equation .529 16.1 Mihăilescu's First Two Theorems .529 16.1.1 The First Theorem: Double Wieferich Pairs .530 16.1.2 The Equation (χΡ - \)/{x - 1) = pyl .532 16.1.3 Mihăilescu's Second Theorem: ρ \| h~ and q \ h~ .536 16.2 The + and - Subspaces and the Group S .537 16.2.1 The + and - Subspaces .538 16.2.2 The Group S .540 16.3 Mihăilescu's Third Theorem: p < Aq2 and q < 4p2 .542 16.4 Mihăilescu's Fourth Theorem: ρ ξ 1 (mod q) or q = 1 (mod p) 547 16.4.1 Preliminaries on Commutative Algebra .547 16.4.2 Preliminaries on the Plus Part .549 16.4.3 Cyclotomic Units and Thaine's Theorem .552 16.4.4 Preliminaries on Power Series .554 16.4.5 Proof of Mihăilescu's Fourth Theorem .557 16.4.6 Conclusion: Proof of Catalan's Conjecture .560 Bibliography .561 Index of Notation .571 Index of Names .579 General Index .585
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spelling	Cohen, Henri 1947- Verfasser (DE-588)1018621717 aut Number theory 2 Analytic and modern tools Henri Cohen New York, NY Springer 2007 XXIII, 596 Seiten txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 240 Graduate texts in mathematics ... (DE-604)BV022220354 2 Erscheint auch als Online-Ausgabe 978-0-387-49894-2 Graduate texts in mathematics 240 (DE-604)BV000000067 240 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015431707&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
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title	Number theory
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title_full	Number theory 2 Analytic and modern tools Henri Cohen
title_fullStr	Number theory 2 Analytic and modern tools Henri Cohen
title_full_unstemmed	Number theory 2 Analytic and modern tools Henri Cohen
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