导语

  

内容提要

  

    《数论》分为2卷，是Springer“数学研究生教材”丛书之239和240卷，是一套面向研究生的数论教程，主旨是全面介绍丢番图方程的解，包括多项式方程、有理数和代数数论等，其中特别强调了算术代数几何的现代理论。全书各章共有530例习题，部分习题有提示。
    本书是其中的第2卷，由H.科恩著。共分2部分8章，内容包括伯努利多项式与伽玛函数、Dirichlet级数和L-函数、p-adicγ和l-函数、线性形式在对数中的应用、高亏格曲线上的有理点等。

Preface
Part III. Analytic Tools
  9. Bernoulli Polynomials and the Gamma Function
    9.1  Bernoulli Numbers and Polynomials
      9.1.1  Generating Functions for Bernoulli Polynomials
      9.1.2  Further Recurrences for Bernoulli Polynomials
      9.1.3  Computing a Single Bernoulli Number
      9.1.4  Bernoulli Polynomials and Fourier Series
    9.2  Analytic Applications of Bernoulli Polynomials
      9.2.1  Asymptotic Expansions
      9.2.2  The Euler-MacLaurin Summation Formula
      9.2.3  The Remainder Term and the Constant Term
      9.2.4  Euler-MacLaurin and the Laplace Transform
      9.2.5  Basic Applications of the Euler-MacLaurin Formula
    9.3  Applications to Numerical Integration
      9.3.1  Standard Euler-MacLaurin Numerical Integration
      9.3.2  The Basic Tanh-Sinh Numerical Integration Method
      9.3.3  General Doubly Exponential Numerical Integration
    9.4  x-Bernoulli Numbers, Polynomials, and Functions
      9.4.1  x-Bernoulli Numbers and Polynomials
      9.4.2  x-Bernoulli Functions
      9.4.3  The x-Euler-MacLaurin Summation Formula
    9.5  Arithmetic Properties of Bernoulli Numbers
      9.5.1  x-Power Sums
      9.5.2  The Generalized Clausen-von Staudt Congruence
      9.5.3  The Voronoi Congruence
      9.5.4  The Kummer Congruences
      9.5.5  The Almkvist-Meurman Theorem
    9.6  The Real and Complex Gamma Functions
      9.6.1  The Hurwitz Zeta Function
      9.6.2  Definition of the Gamma Function
      9.6.3  Preliminary Results for the Study of r(s)
      9.6.4  Properties of the Gamma Function
      9.6.5  Specific Properties of the Function w(s)
      9.6.6  Fourier Expansions of S(s,x) and log(F(x))
    9.7  Integral Transforms
      9.7.1  Generalities on Integral Transforms
      9.7.2  The Fourier Transform
      9.7.3  The Mellin Transform
      9.7.4  The Laplace Transform
    9.8  Bessel Functions
      9.8.1  Definitions
      9.8.2  Integral Representations and Applications
    9.9  Exercises for Chapter 9
  10. Dirichlet Series and L-Functions
    10.1  Arithmetic Functions and Dirichlet Series
      10.1.1  Operations on Arithmetic Functions
      10.1.2  Multiplicative Functions
      10.1.3  Some Classical Arithmetical Functions
      10.1.4  Numerical Dirichlet Series
    10.2  The Analytic Theory of L-Series
      10.2.1  Simple Approaches to Analytic Continuation
      10.2.2  The Use of the Hurwitz Zeta Function S(s, x)
      10.2.3  The Functional Equation for the Theta Function
      10.2.4  The Functional Equation for Dirichlet L-Functions
      10.2.5  Generalized Poisson Summation Formulas
      10.2.6  Voronoi's Error Term in the Circle Problem
    10.3  Special Values of Dirichlet L-Functions
      10.3.1  Basic Results on Special Values
      10.3.2  Special Values of L-Functions and Modular Forms
      10.3.3  The P61ya-Vinogradov Inequality
      10.3.4  Bounds and Averages for L(x, 1)
      10.3.5  Expansions of ((s) Around s = k C Z < 1
      10.3.6  Numerical Computation of Euler Products and Sums
    10.4  Epstein Zeta Functions
      10.4.1  The Nonholomorphic Eisenstein Series G(r, s)
      10.4.2  The Kronecker Limit Formula
    10.5  Dirichlet Series Linked to Number Fields
      10.5.1  The Dedekind Zeta Function Sk(s)
      10.5.2  The Dedekind Zeta Function of Quadratic Fields
      10.5.3  Applications of the Kronecker Limit Formula
      10.5.4  The Dedekind Zeta Function of Cyclotomic Fields
      10.5.5  The Nonvanishing of L(x, 1)
      10.5.6  Application to Primes in Arithmetic Progression
      10.5.7  Conjectures on Dirichlet L-Functions
    10.6  Science Fiction on L-Functions
      10.6.1  Local L-Functions
      10.6.2  Global L-Functions
    10.7  The Prime Number Theorem
      10.7.1  Estimates for S(s)
      10.7.2  Newman's Proof
      10.7.3  Iwaniec's Proof
    10.8  Exercises for Chapter 10
  11. p-adic Gamma and L-Functions
    11.1  Generalities on p-adic Functions
      11.1.1  Methods for Constructing p-adic Functions
      11.1.2  A Brief Study of Volkenborn Integrals
    11.2  The p-adic Hurwitz Zeta Functions
      11.2.1  Teichmfiller Extensions and Characters on Zv
      11.2.2  The p-adic Hurwitz Zeta Function for x E CZp
      11.2.3  The Function Sp(s, x) Around s = 1
      11.2.4  The p-adic Hurwitz Zeta Function for x E Zp
    11.3  p-adic L-Functions
      11.3.1  Dirichlet Characters in the p-adic Context
      11.3.2  Definition and Basic Properties of p-adic L-Functions
      11.3.3  p-adic L-Functions at Positive Integers
      11.3.4  x-Power Sums Involving p-adic Logarithms
      11.3.5  The Function Lp(x, s) Around s = 1
    11.4  Applications of p-adic L-Functions
      11.4.1  Integrality and Parity of L-Function Values
      11.4.2  Bernoulli Numbers and Regular Primes
      11.4.3  Strengthening of the Almkvist-Meurman Theorem
    11.5  p-adic Log Gamma Functions
      11.5.1  Diamond's p-adic Log Gamma Function
      11.5.2  Morita's p-adic Log Gamma Function
      11.5.3  Computation of some p-adic Logarithms
      11.5.4  Computation of Limits of some Logarithmic Sums
      11.5.5  Explicit Formulas for Cp(r/m) and Cv(x, r/m)
      11.5.6  Application to the Value of Lp(x, 1)
    11.6  Morita's p-adic Gamma Function
      11.6.1  Introduction
      11.6.2  Definitions and Basic Results
      11.6.3  Main Properties of the p-adic Gamma Function
      11.6.4  Mahler-Dwork Expansions Linked to Fp(x)
      11.6.5  Power Series Expansions Linked to Fp(x)
      11.6.6  The Jacobstahl-Kazandzidis Congruence
    11.7  The Gross-Koblitz Formula and Applications
      11.7.1  Statement and Proof of the Gross-Koblitz Formula
      11.7.2  Application to Lp(x,O)
      11.7.3  Application to the Stickelberger Congruence
      11.7.4  Application to the Hasse-Davenport Product Relation
    11.8  Exercises for Chapter 11
Part IV. Modern Tools
  12. Applications of Linear Forms in Logarithms
    12.1  Introduction
      12.1.1  Lower Bounds
      12.1.2  Applications to Diophantine Equations and Problems
      12.1.3  A List of Applications
    12.2  A Lower Bound for 12m - 3hi
    12.3  Lower Bounds for the Trace of cn
    12.4  Pure Powers in Binary Recurrent Sequences
    12.5  Greatest Prime Factors of Terms of Some Recurrent Se quences
    12.6  Greatest Prime Factors of Values of Integer Polynomials
    12.7  The Diophantine Equation axn - byn = c
    12.8  Simultaneous Pell Equations
      12.8.1  General Strategy
      12.8.2  An Example in Detail
      12.8.3  A General Algorithm
    12.9  Catalan's Equation
    12.10  Thue Equations
      12.10.1  The Main Theorem
      12.10.2  Algorithmic Aspects
    12.11  Other Classical Diophantine Equations
    12.12  A Few Words on the Non-Archimedean Case
  13. Rational Points on Higher-Genus Curves
    13.1  Introduction
    13.2  The Jacobian
      13.2.1  Functions on Curves
      13.2.2  Divisors
      13.2.3  Rational Divisors
      13.2.4  The Group Law: Cantor's Algorithm
      13.2.5  The Group Law: The Geometric Point of View
    13.3  Rational Points on Hyperelliptic Curves
      13.3.1  The Method of Demtyanenko-Manin
      13.3.2  The Method of Chabauty-Coleman
      13.3.3  Explicit Chabauty According to Flynn
      13.3.4  When Chabauty Fails
      13.3.5  Elliptic Curve Chabauty
      13.3.6  A Complete Example
  14. The Super-Fermat Equation
    14.1  Preliminary Reductions
    14.2  The Dihedral Cases (2, 2, r)
      14.2.1  The Equation x2 - y2 = zr
      14.2.2  The Equation x2 + y2 = zr
      14.2.3  The Equations x2 + 3y2 = z3 and X2 + 3y2 = 4Z3
    14.3  The Tetrahedral Case (2, 3, 3)
      14.3.1  The Equation x3 + y3 = z2
      14.3.2  The Equation x3 + y3 = 2z2
      14.3.3  The Equation x3 - 2y3 = z2
    14.4  The Octa.hedral Case (2, 3, 4)
      14.4.1  The Equation x2 - y4 = z3
      14.4.2  The Equation x2 + y4 = z3
    14.5  Invariants, Covariants, and Dessins d'Enfants
      14.5.1  Dessins d'Enfants, Klein Forms, and Covariants
      14.5.2  The Icosahedral Case (2, 3, 5)
    14.6  The Parabolic and Hyperbolic Cases
      14.6.1  The Parabolic Case
      14.6.2  General Results in the Hyperbolic Case
      14.6.3  The Equations x4 + y4 = z3
      14.6.4  The Equation x4 + y4 = z5
      14.6.5  The Equation x6 - y4 = z2
      14.6.6  The Equation x4 - y6 = z2
      14.6.7  The Equation x6 + y4 = z2
      14.6.8  Further Results
    14.7  Applications of Mason's Theorem
      14.7.1  Mason's Theorem
      14.7.2  Applications
    14.8  Exercises for Chapter 14
  15. The Modular Approach to Diophantine Equations
    15.1  Newforms
      15.1.1  Introduction and Necessary Software Tools
      15.1.2  Newforms
      15.1.3  Rational Newforms and Elliptic Curves
    15.2  Ribet's Level-Lowering Theorem
      15.2.1  Definition of "Arises From"
      15.2.2  Ribet's Level-Lowering Theorem
      15.2.3  Absence of Isogenies
      15.2.4  How to use Ribet's Theorem
    15.3  Fermat's Last Theorem and Similar Equations
      15.3.1  A Generalization of FLT
      15.3.2  E Arises from a Curve with Complex Multiplication
   15.3.3 End of the Proof of Theorem 15.3.1
      15.3.4  The Equation x2 = yP + 2rZp for p > 7 and r > 2
      15.3.5  The Equation x2 = yP + zp for p > 7
    15.4  An Occasional Bound for the Exponent
    15.5  An Example of Serre-Mazur-Kraus
    15.6  The Method of Kraus
    15.7  "Predicting Exponents of Constants"
      15.7.1  The Diophantine Equation x2 - 2 = yP
      15.7.2  Application to the SMK Equation
    15.8  Recipes for Some Ternary Diophantine Equations
      15.8.1  Recipes for Signature (p, p, p)
      15.8.2  Recipes for Signature (p, p, 2)
      15.8.3  Recipes for Signature (p, p, 3)
  16. Catalan's Equation
    16.1  Mihailescu's First Two Theorems
      16.1.1  The First Theorem: Double Wieferich Pairs
      16.1.2  The Equation (xp - 1)/(x - 1) = pyq
      16.1.3  Mihailescu's Second Theorem: p | hq and q | hp
    16.2  The + and - Subspaces and the Group S
      16.2.1  The + and - Subspaces
      16.2.2  The Group S
    16.3  Mihailescu's Third Theorem: p < 4q2 and q < 4p2
    16.4  Mihailescu's Fourth Theorem: p = 1 (mod q) or q = 1 (mod p)
      16.4.1  Preliminaries on Commutative Algebra
      16.4.2  Preliminaries on the Plus Part
      16.4.3  Cyclotomic Units and Thaine's Theorem
      16.4.4  Preliminaries on Power Series
      16.4.5  Proof of Mihailescu's Fourth Theorem
      16.4.6  Conclusion: Proof of Catalan's Conjecture
Bibliography
Index of Notation
Index of Names
General Index