Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most recently the ABC conjecture. This monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The authors provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and provide a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained, with proofs given in full detail.
Author(s): Bombieri E., Gubler W.
Series: New Mathematical Monographs
Publisher: CUP
Year: 2006
Language: English
Pages: 669
Tags: Математика;Высшая геометрия;Алгебраическая геометрия;
Contents......Page 6
Preface......Page 12
Terminology......Page 16
1.2. Absolute values......Page 18
1.3. Finite-dimensional extensions......Page 22
1.4. The product formula......Page 26
1.5. Heights in projective and affine space......Page 32
1.6. Heights of polynomials......Page 38
1.7. Lower bounds for norms of products of polynomials......Page 46
1.8. Bibliographical notes......Page 50
2.1. Introduction......Page 51
2.2. Local heights......Page 52
2.3. Global heights......Page 56
2.4. Weil heights......Page 59
2.5. Explicit bounds for Weil heights......Page 62
2.6. Bounded subsets......Page 71
2.7. Metrized line bundles and local heights......Page 74
2.8. Heights on Grassmannians......Page 83
2.9. Siegel s lemma......Page 89
2.10. Bibliographical notes......Page 97
3.2. Subgroups and lattices......Page 99
3.3. Subvarieties and maximal subgroups......Page 105
3.4. Bibliographical notes......Page 109
4.2. Zhang's theorem......Page 110
4.3. The equidistribution theorem......Page 118
4.4. Dobrowolski s theorem......Page 124
4.5. Remarks on the Northcott property......Page 134
4.6. Remarks on the Bogomolov property......Page 137
4.7. Bibliographical notes......Page 140
5.1. Introduction......Page 142
5.2. The number of solutions of the unit equation......Page 143
5.3. Applications......Page 157
5.4. Effective methods......Page 163
5.5. Bibliographical notes......Page 166
6.1. Introduction......Page 167
6.2. Roth s theorem......Page 169
6.3. Preliminary lemmas......Page 173
6.4. Proof of Roth s theorem......Page 180
6.5. Further results......Page 187
6.6. Bibliographical notes......Page 191
7.1. Introduction......Page 193
7.2. The subspace theorem......Page 194
7.3. Applications......Page 198
7.4. The generalized unit equation......Page 203
7.5. Proof of the subspace theorem......Page 214
7.6. Further results: the product theorem......Page 243
7.7. The absolute subspace theorem and the Faltings?Wustholz theorem......Page 244
7.8. Bibliographical notes......Page 247
8.1. Introduction......Page 248
8.2. Group varieties......Page 249
8.3. Elliptic curves......Page 257
8.4. The Picard variety......Page 263
8.5. The theorem of the square and the dual abelian variety......Page 269
8.6. The theorem of the cube......Page 274
8.7. The isogeny multiplication by......Page 280
8.8. Characterization of odd elements in the Picard group......Page 282
8.9. Decomposition into simple abelian varieties......Page 284
8.10. Curves and Jacobians......Page 285
8.11. Bibliographical notes......Page 299
9.1. Introduction......Page 300
9.2. N´eron-Tate heights......Page 301
9.3. The associated bilinear form......Page 306
9.4. N´eron-Tate heights on Jacobians......Page 311
9.5. The N´eron symbol......Page 318
9.6. Hilbert’s irreducibility theorem......Page 331
9.7. Bibliographical notes......Page 343
10.1. Introduction......Page 345
10.2. The weak Mordell?Weil theorem for elliptic curves......Page 346
10.3. The Chevalley?Weil theorem......Page 352
10.4. The weak Mordell?Weil theorem for abelian varieties......Page 358
10.5. Kummer theory and Galois cohomology......Page 361
10.6. The Mordell?Weil theorem......Page 366
10.7. Bibliographical notes......Page 368
11.1. Introduction......Page 369
11.2. The Vojta divisor......Page 373
11.3. Mumford s method and an upper bound for the height......Page 376
11.4. The local Eisenstein theorem......Page 377
11.5. Power series, norms, and the local Eisenstein theorem......Page 379
11.6. A lower bound for the height......Page 388
11.7. Construction of a Vojta divisor of small height......Page 393
11.8. Application of Roth s lemma......Page 398
11.9. Proof of Faltings s theorem......Page 404
11.10. Some further developments......Page 408
11.11. Bibliographical notes......Page 417
12.1. Introduction......Page 418
12.2. The abc-conjecture......Page 419
12.3. Bely?’s theorem......Page 428
12.4. Examples......Page 433
12.5. Equivalent conjectures......Page 441
12.6. The generalized Fermat equation......Page 452
12.7. Bibliographical notes......Page 459
13.2. Nevanlinna theory in one variable......Page 461
13.3. Variations on a theme: the Ahlfors?Shimizu characteristic......Page 474
13.4. Holomorphic curves in Nevanlinna theory......Page 482
13.5. Bibliographical notes......Page 494
14.1. Introduction......Page 496
14.2. The Vojta dictionary......Page 497
14.3. Vojta s conjectures......Page 500
14.4. A general abc-conjecture......Page 505
14.5. The abc-theorem for function fields......Page 515
14.6. Bibliographical notes......Page 530
A.2. Affine varieties......Page 531
A.3. Topology and sheaves......Page 535
A.4. Varieties......Page 538
A.5. Vector bundles......Page 542
A.6. Projective varieties......Page 547
A.7. Smooth varieties......Page 553
A.8. Divisors......Page 561
A.9. Intersection theory of divisors......Page 568
A.10. Cohomology of sheaves......Page 580
A.11. Rational maps......Page 591
A.12. Properties of morphisms......Page 594
A.13. Curves and surfaces......Page 598
A.14. Connexion to complex manifolds......Page 600
B.1. Discriminants......Page 603
B.2. Unramified field extensions......Page 608
B.3. Unramified morphisms......Page 615
B.4. The rami“cation divisor......Page 616
C.1. Adeles......Page 619
C.2. Minkowski s second theorem......Page 625
C.3. Cube slicing......Page 632
Glossary of Notation......Page 652
Index......Page 660
References......Page 637
