Number Theory

Table of Contentsxv8.2.3 The Fundamental 2-Descent Map . . . . . . . . . . . . . . . . . . . 5358.2.4 Practical Use of 2-Descent with 2-Isogenies . . . . . . . . . . 5388.2.5 Examples of 2-Descent with 2-Isogenies . . . . . . . . . . . . . 5428.2.6 An Example of Second Descent . . . . . . . . . . . . . . . . . . . . 5468.3 Description of General 2-Descent . . . . . . . . . . . . . . . . . . . . . . . . . 5488.3.1 The Fundamental 2-Descent Map . . . . . . . . . . . . . . . . . . . 5488.3.2 The T-Selmer Group of a Number Field . . . . . . . . . . . . . 5508.3.3 Description of the Image of α . . . . . . . . . . . . . . . . . . . . . . 5528.3.4 Practical Use of 2-Descent in the General Case . . . . . . . 5548.3.5 Examples of General 2-Descent. . . . . . . . . . . . . . . . . . . . . 5558.4 Description of 3-Descent with Rational 3-Torsion Subgroup . . 5568.4.1 Rational 3-Torsion Subgroups . . . . . . . . . . . . . . . . . . . . . . 5578.4.2 The Fundamental 3-Isogeny. . . . . . . . . . . . . . . . . . . . . . . . 5588.4.3 Description of the Image of φ . . . . . . . . . . . . . . . . . . . . . . 5608.4.4 The Fundamental 3-Descent Map . . . . . . . . . . . . . . . . . . . 5638.5 The Use of L(E, s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5648.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5648.5.2 The Case of Complex Multiplication . . . . . . . . . . . . . . . . 5658.5.3 Numerical Computation of L (r) (E, 1) . . . . . . . . . . . . . . . 5728.5.4 Computation of Γ r (1, x) for Small x . . . . . . . . . . . . . . . . 5758.5.5 Computation of Γ r (1, x) for Large x . . . . . . . . . . . . . . . . 5808.5.6 The Famous Curve y 2 + y = x 3 − 7x + 6 . . . . . . . . . . . . 5828.6 The Heegner Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5848.6.1 Introduction and the Modular Parametrization . . . . . . . 5848.6.2 Heegner Points and Complex Multiplication . . . . . . . . . 5868.6.3 Use of the Theorem of Gross–Zagier . . . . . . . . . . . . . . . . 5898.6.4 Practical Use of the Heegner Point Method . . . . . . . . . . 5918.6.5 Improvements to the Basic Algorithm, in Brief . . . . . . . 5958.6.6 A Complete Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5988.7 Computation of Integral Points. . . . . . . . . . . . . . . . . . . . . . . . . . . 5998.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6008.7.2 An Upper Bound for the Elliptic Logarithm on E(Z) . 6008.7.3 Lower Bounds for Linear Forms in Elliptic Logarithms 6038.7.4 A Complete Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6058.8 Exercises for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607Part III. Analytic Methods9. Bernoulli Polynomials and the Gamma Function . . . . . . . . . . 6179.1 Bernoulli Numbers and Polynomials . . . . . . . . . . . . . . . . . . . . . . 6179.1.1 Generating Functions for Bernoulli Polynomials . . . . . . 6179.1.2 Further Recursions for Bernoulli Polynomials . . . . . . . . 6249.1.3 Computing a Single Bernoulli Number . . . . . . . . . . . . . . 6299.1.4 Bernoulli Polynomials and Fourier Series . . . . . . . . . . . . 630