Heights in diophantine geometry / Enrico Bombieri, Walter Gubler.

Format:

Book

Author/Creator:

Bombieri, Enrico, 1940-

Contributor:

Gubler, Walter.

Class of 1932 Fund.

Series:

New mathematical monographs ; 4.

New mathematical monographs ; 4

Language:

English

Subjects (All):

Arithmetical algebraic geometry.

Physical Description:

xvi, 652 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge : Cambridge University Press, 2006.

Summary:

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 a modern approach via arithmetic geometry. The authors aim to provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained (there are appendices on algebraic geometry, ramification and the geometry of numbers), with proofs given in full detail. Many results appear here for the first time.

The first half of the book is devoted to the general theory of heights and its applications including a complete, detailed proof of the celebrated subspace theorem of W.M. Schmidt.

The second part deals with Abelian varieties, the Mordell-Weil theorem and Faltings' proof of the Mordell conjecture, ending with a self-contained exposition of Nevanlinna theory and the related famous conjectures of Vojta. The book concludes with a comprehensive bibliography. It is destined to be a definitive reference on modern diophantine geometry, bringing a new standard of rigor and elegance to the field.

Contents:

1 Heights 1

1.2 Absolute values 1

1.3 Finite-dimensional extensions 5

1.4 The product formula 9

1.5 Heights in projective and affine space 15

1.6 Heights of polynomials 21

1.7 Lower bounds for norms of products of polynomials 29

2 Weil heights 34

2.2 Local heights 35

2.3 Global heights 39

2.4 Weil heights 42

2.5 Explicit bounds for Weil heights 45

2.6 Bounded subsets 54

2.7 Metrized line bundles and local heights 57

2.8 Heights on Grassmannians 66

2.9 Siegel's lemma 72

3 Linear tori 82

3.2 Subgroups and lattices 82

3.3 Subvarieties and maximal subgroups 88

4 Small points 93

4.2 Zhang's theorem 93

4.3 The equidistribution theorem 101

4.4 Dobrowolski's theorem 107

4.5 Remarks on the Northcott property 117

4.6 Remarks on the Bogomolov property 120

5 The unit equation 125

5.2 The number of solutions of the unit equation 126

5.3 Applications 140

5.4 Effective methods 146

6 Roth's theorem 150

6.2 Roth's theorem 152

6.3 Preliminary lemmas 156

6.4 Proof of Roth's theorem 163

6.5 Further results 170

7 The subspace theorem 176

7.2 The subspace theorem 177

7.3 Applications 181

7.4 The generalized unit equation 186

7.5 Proof of the subspace theorem 197

7.6 Further results: the product theorem 226

7.7 The absolute subspace theorem and the Faltings-Wustholz theorem 227

8 Abelian varieties 231

8.2 Group varieties 232

8.3 Elliptic curves 240

8.4 The Picard variety 246

8.5 The theorem of the square and the dual abelian variety 252

8.6 The theorem of the cube 257

8.7 The isogeny multiplication by n 263

8.8 Characterization of odd elements in the Picard group 265

8.9 Decomposition into simple abelian varieties 267

8.10 Curves and Jacobians 268

9 Neron-Tate heights 283

9.2 Neron-Tate heights 284

9.3 The associated bilinear form 289

9.4 Neron-Tate heights on Jacobians 294

9.5 The Neron symbol 301

9.6 Hilbert's irreducibility theorem 314

10 The Mordell-Weil theorem 328

10.2 The weak Mordell-Weil theorem for elliptic curves 329

10.3 The Chevalley-Weil theorem 335

10.4 The weak Mordell-Weil theorem for abelian varieties 341

10.5 Kummer theory and Galois cohomology 344

10.6 The Mordell-Weil theorem 349

11 Faltings's theorem 352

11.2 The Vojta divisor 356

11.3 Mumford's method and an upper bound for the height 359

11.4 The local Eisenstein theorem 360

11.5 Power series, norms, and the local Eisenstein theorem 362

11.6 A lower bound for the height 371

11.7 Construction of a Vojta divisor of small height 376

11.8 Application of Roth's lemma 381

11.9 Proof of Faltings's theorem 387

11.10 Some further developments 391

12 The abc-conjecture 401

12.2 The abc-conjecture 402

12.3 Belyi's theorem 411

12.5 Equivalent conjectures 424

12.6 The generalized Fermat equation 435

13 Nevanlinna theory 444

13.2 Nevanlinna theory in one variable 444

13.3 Variations on a theme: the Ahlfors-Shimizu characteristic 457

13.4 Holomorphic curves in Nevanlinna theory 465

14 The Vojta conjectures 479

14.2 The Vojta dictionary 480

14.3 Vojta's conjectures 483

14.4 A general abc-conjecture 488

14.5 The abc-theorem for function fields 498

Appendix A Algebraic geometry 514

A.2 Affine varieties 514

A.3 Topology and sheaves 518

A.4 Varieties 521

A.5 Vector bundles 525

A.6 Projective varieties 530

A.7 Smooth varieties 536

A.8 Divisors 544

A.9 Intersection theory of divisors 551

A.10 Cohomology of sheaves 563

A.11 Rational maps 574

A.12 Properties of morphisms 577

A.13 Curves and surfaces 581

A.14 Connexion to complex manifolds 583

Appendix B Ramification 586

B.1 Discriminants 586

B.2 Unramified field extensions 591

B.3 Unramified morphisms 598

B.4 The ramification divisor 599

Appendix C Geometry of numbers 602

C.1 Adeles 602

C.2 Minkowski's second theorem 608

C.3 Cube slicing 615.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1932 Fund.

ISBN:

0521846153

OCLC:

62132904

Publisher Number:

9780521846158