Preface |
Elliptic curves / 1: |
Normal forms / 1.1: |
The addition law / 1.2: |
Multiplication formulas / 1.3: |
Factorization and primality test / 1.4: |
Isogenies and endomorphisms of elliptic curves / 1.5: |
Exercises / 1.6: |
Elliptic curves over the complex numbers / 2: |
Lattices / 2.1: |
Weierstrass [Weierstrass p]-function / 2.2: |
Periods of elliptic curves / 2.3: |
Complex multiplication / 2.4: |
Elliptic curves over finite fields / 2.5: |
Frobenius endomorphism and supersingular curves / 3.1: |
Computing the number of points / 3.2: |
Construction of elliptic curves with given group order / 3.3: |
Elliptic curves in cryptography / 3.4: |
The discrete logarithm problem on elliptic curves / 3.5: |
Elliptic curves over local fields / 3.6: |
Reduction / 4.1: |
The filtration / 4.2: |
The theorem of Nagell, Lutz, and Cassels / 4.3: |
The Mordell-Weil theorem and heights / 4.4: |
Theorem of Mordell and Weil / 5.1: |
Heights / 5.2: |
Computation of the heights / 5.3: |
Points of bounded height / 5.4: |
The differences between the heights / 5.5: |
Torsion group / 5.6: |
Structure of the torsion group / 6.1: |
Elliptic curves with integral j-invariant / 6.2: |
Computation of the torsion group / 6.3: |
Examples / 6.6: |
The rank / 6.7: |
L-series / 7.1: |
The coefficients of the L-series / 7.2: |
Continuation of the L-series / 7.3: |
Conjectures concerning the rank / 7.4: |
The Selmer and the Tate-Shafarevich group / 7.5: |
2-descent / 7.6: |
The rank in field extensions / 7.7: |
Basis / 7.8: |
Linearly independent points / 8.1: |
Computation of a basis / 8.2: |
Heegner point method / 8.3: |
S-integral points / 8.5: |
Overview / 9.1: |
Elliptic logarithms / 9.2: |
S-integral points over Q / 9.3: |
Proof of the theorem / 9.4: |
Example / 9.5: |
Algorithmic theory of diophantine equations / 9.6: |
Hilbert's 10th problem / A.1: |
Introduction to Baker's method / A.2: |
S-unit equations / A.3: |
Thue equations / A.4: |
Small collection of other results / A.5: |
Lower bounds for linear forms in logarithms / A.6: |
LLL-algorithm / A.7: |
Reduction of the large bound / A.8: |
Multiquadratic number fields / B: |
Multiquadratic fields and Galois groups / B.1: |
Discriminants / B.2: |
Integral Bases / B.3: |
Decomposition Law / B.4: |
Biquadratic number fields / B.5: |
Totally real and totally complex biquadratic fields / B.6: |
Bibliography / B.7: |
Index |