Heights in diophantine geometry:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Edition: | Reprinted with corr. |
| Series: | New mathematical monographs
4 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 652 S. |
| ISBN: | 9780521712293 9780521846158 |
Staff View
MARC
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| 100 | 1 | |a Bombieri, Enrico |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Heights in diophantine geometry |c Enrico Bombieri ; Walter Gubler |
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| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2007 | |
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Record in the Search Index
| _version_ | 1804137269680930816 |
|---|---|
| adam_text | Contents
Preface
page
xi
Terminology
XV
1.
Heights
1
1.1.
Introduction
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
1.8.
Bibliographical notes
33
2.
Weil heights
34
2.1.
Introduction
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.
Siegeľs
lemma
72
2.10.
Bibliographical notes
80
3.
Linear tori
82
3.1.
Introduction
82
3.2.
Subgroups and lattices
82
3.3.
Subvarieties and maximal subgroups
88
3.4. Biblioeraühical
notes
92
vi
Contents
4.
Small
points
93
4.1.
Introduction
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
4.7.
Bibliographical notes
123
5.
The unit equation
125
5.1.
Introduction
125
5.2.
The number of solutions of the unit equation
126
5.3.
Applications
140
5.4.
Effective methods
146
5.5.
Bibliographical notes
149
6.
Roth s theorem
150
6.1.
Introduction
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
6.6.
Bibliographical notes
174
7.
The subspace theorem
176
7.1.
Introduction
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-Wiistholz
theorem
227
7.8.
Bibliographical notes
230
8.
Abelian varieties
231
8.1.
Introduction
231
8.2.
Group varieties
232
8.3.
Elliptic curves
240
8.4.
The
Picard
variety
246
Contents
vii
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
η
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
8.11.
Bibliographical notes
282
9.
Néron-Tate
heights
283
9.1.
Introduction
283
9.2.
Néron-Tate
heights
284
9.3.
The associated bilinear form
289
9.4.
Néron-Tate
heights on Jacobians
294
9.5.
The
Néron
symbol
301
9.6.
Hubert s irreducibility theorem
314
9.7.
Bibliographical notes
326
10.
The Mordell-Weil theorem
328
10.1.
Introduction
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
10.7.
Bibliographical notes
351
11.
Faltings s theorem
352
11.1.
Introduction
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
11.11.
Bibliographical notes
400
viii Contents
12.
The abc
-conjecture
401
12.1.
Introduction
401
12.2.
The
abc
-conjecture
402
12.3.
Belyï s
theorem
411
12.4.
Examples
416
12.5.
Equivalent conjectures
424
12.6.
The generalized
Fermat
equation
435
12.7.
Bibliographical notes
442
13.
Nevanlinna theory
444
13.1.
Introduction
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
13.5.
Bibliographical notes
477
14.
The
Vojta
conjectures
479
14.1.
Introduction
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
14.6.
Bibliographical notes
513
Appendix A. Algebraic geometry
514
514
514
518
521
525
530
536
544
551
563
574
577
581
583
A.I.
Introduction
A.2.
Affine
varieties
A.3.
Topology and sheaves
A.4.
Varieties
A.5.
Vector bundles
A.6.
Projective
varieties
A.7.
Smooth varieties
A.8.
Divisors
A.9.
Intersection theory of divisors
АЛО.
Cohomology of sheaves
A.ll.
Rational maps
A.12.
Properties of morphisms
A.
13.
Curves and surfaces
A.
14.
Connexion to complex manifolds
Contents ix
Appendix
В.
Ramification
586
B.I. 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.I. Adeles
602
C.2. Minkowski s second theorem
608
C.3. Cube slicing
615
References
620
Glossary of notation
635
Index
643
|
| adam_txt |
Contents
Preface
page
xi
Terminology
XV
1.
Heights
1
1.1.
Introduction
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
1.8.
Bibliographical notes
33
2.
Weil heights
34
2.1.
Introduction
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.
Siegeľs
lemma
72
2.10.
Bibliographical notes
80
3.
Linear tori
82
3.1.
Introduction
82
3.2.
Subgroups and lattices
82
3.3.
Subvarieties and maximal subgroups
88
3.4. Biblioeraühical
notes
92
vi
Contents
4.
Small
points
93
4.1.
Introduction
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
4.7.
Bibliographical notes
123
5.
The unit equation
125
5.1.
Introduction
125
5.2.
The number of solutions of the unit equation
126
5.3.
Applications
140
5.4.
Effective methods
146
5.5.
Bibliographical notes
149
6.
Roth's theorem
150
6.1.
Introduction
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
6.6.
Bibliographical notes
174
7.
The subspace theorem
176
7.1.
Introduction
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-Wiistholz
theorem
227
7.8.
Bibliographical notes
230
8.
Abelian varieties
231
8.1.
Introduction
231
8.2.
Group varieties
232
8.3.
Elliptic curves
240
8.4.
The
Picard
variety
246
Contents
vii
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
η
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
8.11.
Bibliographical notes
282
9.
Néron-Tate
heights
283
9.1.
Introduction
283
9.2.
Néron-Tate
heights
284
9.3.
The associated bilinear form
289
9.4.
Néron-Tate
heights on Jacobians
294
9.5.
The
Néron
symbol
301
9.6.
Hubert's irreducibility theorem
314
9.7.
Bibliographical notes
326
10.
The Mordell-Weil theorem
328
10.1.
Introduction
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
10.7.
Bibliographical notes
351
11.
Faltings's theorem
352
11.1.
Introduction
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
11.11.
Bibliographical notes
400
viii Contents
12.
The abc
-conjecture
401
12.1.
Introduction
401
12.2.
The
abc
-conjecture
402
12.3.
Belyï's
theorem
411
12.4.
Examples
416
12.5.
Equivalent conjectures
424
12.6.
The generalized
Fermat
equation
435
12.7.
Bibliographical notes
442
13.
Nevanlinna theory
444
13.1.
Introduction
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
13.5.
Bibliographical notes
477
14.
The
Vojta
conjectures
479
14.1.
Introduction
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
14.6.
Bibliographical notes
513
Appendix A. Algebraic geometry
514
514
514
518
521
525
530
536
544
551
563
574
577
581
583
A.I.
Introduction
A.2.
Affine
varieties
A.3.
Topology and sheaves
A.4.
Varieties
A.5.
Vector bundles
A.6.
Projective
varieties
A.7.
Smooth varieties
A.8.
Divisors
A.9.
Intersection theory of divisors
АЛО.
Cohomology of sheaves
A.ll.
Rational maps
A.12.
Properties of morphisms
A.
13.
Curves and surfaces
A.
14.
Connexion to complex manifolds
Contents ix
Appendix
В.
Ramification
586
B.I. 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.I. Adeles
602
C.2. Minkowski's second theorem
608
C.3. Cube slicing
615
References
620
Glossary of notation
635
Index
643 |
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| id | DE-604.BV023041101 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:20:39Z |
| indexdate | 2024-07-09T21:09:37Z |
| institution | BVB |
| isbn | 9780521712293 9780521846158 |
| language | English |
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| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | New mathematical monographs |
| series2 | New mathematical monographs |
| spelling | Bombieri, Enrico Verfasser aut Heights in diophantine geometry Enrico Bombieri ; Walter Gubler Reprinted with corr. Cambridge [u.a.] Cambridge Univ. Press 2007 XVI, 652 S. txt rdacontent n rdamedia nc rdacarrier New mathematical monographs 4 Arithmetical algebraic geometry Diophantische Geometrie (DE-588)4150021-0 gnd rswk-swf Diophantische Geometrie (DE-588)4150021-0 s b DE-604 Gubler, Walter Verfasser aut New mathematical monographs 4 (DE-604)BV035420183 4 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016244634&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bombieri, Enrico Gubler, Walter Heights in diophantine geometry New mathematical monographs Arithmetical algebraic geometry Diophantische Geometrie (DE-588)4150021-0 gnd |
| subject_GND | (DE-588)4150021-0 |
| title | Heights in diophantine geometry |
| title_auth | Heights in diophantine geometry |
| title_exact_search | Heights in diophantine geometry |
| title_exact_search_txtP | Heights in diophantine geometry |
| title_full | Heights in diophantine geometry Enrico Bombieri ; Walter Gubler |
| title_fullStr | Heights in diophantine geometry Enrico Bombieri ; Walter Gubler |
| title_full_unstemmed | Heights in diophantine geometry Enrico Bombieri ; Walter Gubler |
| title_short | Heights in diophantine geometry |
| title_sort | heights in diophantine geometry |
| topic | Arithmetical algebraic geometry Diophantische Geometrie (DE-588)4150021-0 gnd |
| topic_facet | Arithmetical algebraic geometry Diophantische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016244634&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035420183 |
| work_keys_str_mv | AT bombierienrico heightsindiophantinegeometry AT gublerwalter heightsindiophantinegeometry |