Preface |
|
v | |
|
|
1 | (32) |
|
|
1 | (10) |
|
|
11 | (8) |
|
|
19 | (5) |
|
Factorization and primality test |
|
|
24 | (3) |
|
Isogenies and endomorphisms of elliptic curves |
|
|
27 | (3) |
|
|
30 | (3) |
|
Elliptic curves over the complex numbers |
|
|
33 | (30) |
|
|
33 | (3) |
|
|
36 | (16) |
|
Periods of elliptic curves |
|
|
52 | (3) |
|
|
55 | (7) |
|
|
62 | (1) |
|
Elliptic curves over finite fields |
|
|
63 | (24) |
|
Frobenius endomorphism and supersingular curves |
|
|
63 | (2) |
|
Computing the number of points |
|
|
65 | (9) |
|
Construction of elliptic curves with given group order |
|
|
74 | (5) |
|
Elliptic curves in cryptography |
|
|
79 | (4) |
|
The discrete logarithm problem on elliptic curves |
|
|
83 | (2) |
|
|
85 | (2) |
|
Elliptic curves over local fields |
|
|
87 | (16) |
|
|
87 | (6) |
|
|
93 | (5) |
|
The theorem of Nagell, Lutz, and Cassels |
|
|
98 | (4) |
|
|
102 | (1) |
|
The Mordell--Weil theorem and heights |
|
|
103 | (44) |
|
Theorem of Mordell and Weil |
|
|
103 | (13) |
|
|
116 | (12) |
|
Computation of the heights |
|
|
128 | (5) |
|
|
133 | (3) |
|
The differences between the heights |
|
|
136 | (9) |
|
|
145 | (2) |
|
|
147 | (51) |
|
Structure of the torsion group |
|
|
147 | (4) |
|
Elliptic curves with integral j-invariant |
|
|
151 | (26) |
|
The theorem of Nagell, Lutz, and Cassels |
|
|
177 | (5) |
|
|
182 | (3) |
|
Computation of the torsion group |
|
|
185 | (2) |
|
|
187 | (9) |
|
|
196 | (2) |
|
|
198 | (44) |
|
|
198 | (4) |
|
The coefficients of the L-series |
|
|
202 | (5) |
|
Continuation of the L-series |
|
|
207 | (7) |
|
Conjectures concerning the rank |
|
|
214 | (2) |
|
The Selmer and the Tate--Shafarevich group |
|
|
216 | (12) |
|
|
228 | (5) |
|
The rank in field extensions |
|
|
233 | (7) |
|
|
240 | (2) |
|
|
242 | (21) |
|
Linearly independent points |
|
|
242 | (5) |
|
|
247 | (4) |
|
|
251 | (3) |
|
|
254 | (6) |
|
|
260 | (3) |
|
|
263 | (31) |
|
|
263 | (2) |
|
|
265 | (7) |
|
|
272 | (6) |
|
|
278 | (9) |
|
|
287 | (5) |
|
|
292 | (2) |
|
A Algorithmic theory of diophantine equations |
|
|
294 | (22) |
|
|
294 | (1) |
|
Introduction to Baker's method |
|
|
295 | (3) |
|
|
298 | (5) |
|
|
303 | (2) |
|
Small collection of other results |
|
|
305 | (3) |
|
Lower bounds for linear forms in logarithms |
|
|
308 | (1) |
|
|
309 | (2) |
|
Reduction of the large bound |
|
|
311 | (5) |
|
B Multiquadratic number fields |
|
|
316 | (35) |
|
Multiquadratic fields and Galois groups |
|
|
316 | (1) |
|
|
317 | (4) |
|
|
321 | (3) |
|
|
324 | (6) |
|
Biquadratic number fields |
|
|
330 | (11) |
|
Totally real and totally complex biquadratic fields |
|
|
341 | (8) |
|
|
349 | (2) |
Bibliography |
|
351 | (14) |
Index |
|
365 | |