Fundamentals of diophantine geometry:
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| Previous Title: | Diophantine geometry |
|---|---|
| Main Author: | |
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1983
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Frühere Ausg. u.d.T.: Lang, Serge: Diophantine geometry |
| Physical Description: | XVIII, 370 S. |
| ISBN: | 0387908374 3540908374 9783540908371 9780387908373 |
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MARC
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| 245 | 1 | 0 | |a Fundamentals of diophantine geometry |c Serge Lang |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1983 | |
| 300 | |a XVIII, 370 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 500 | |a Frühere Ausg. u.d.T.: Lang, Serge: Diophantine geometry | ||
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| 650 | 4 | |a Arithmetical algebraic geometry | |
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Record in the Search Index
| _version_ | 1804114750671421440 |
|---|---|
| adam_text | Contents
Acknowledgment
.......................... xv
Some Standard Notation
......................xvii
CHAPTER
1
Absolute Values
.......................... 1
1.
Definitions, dependence and independence
............... 1
2.
Completions
............................ 5
3.
Unramified extensions
........................ 9
4.
Finite extensions
.......................... 12
CHAPTER
2
Proper Sets of Absolute Values.
Divisors and Units
......................... 18
1.
Proper sets of absolute values
..................... 18
2.
Number fields
........................... 19
3.
Divisors on varieties
........................ 21
4.
Divisors on schemes
......................... 24
5.
Afg-divisors and divisor classes
.................... 29
6.
Ideal classes and units in number fields
................. 32
7.
Relative units and divisor classes
................... 41
8.
The Chevalley-Weil theorem
..................... 44
CHAPTER
3
Heights
............................... 50
1.
Definitions
............................. 50
2.
Gauss lemma
........................... 54
3.
Heights in function fields
...................... 62
XU Contents
4.
Heights on abelian groups
...................... 66
5.
Counting points of bounded height
.................. 70
CHAPTER
4
Geometric Properties of Heights
................... 76
1.
Functorial properties
........................ 76
2.
Heights and linear systems
...................... 83
3.
Ample linear systems
........................ 87
4.
Projections on curves
........................ 90
5.
Heights associated with divisor classes
................. 91
CHAPTER
5
Heights on Abelian Varieties
.................... 95
1.
Some linear and quasi-linear algebra
.................. 95
2.
Quadraticity of endomorphisms on divisor classes
............ 99
3.
Quadraticity of the height
...................... 106
4.
Heights and
Poincaré
divisors
.................... 110
5.
Jacobian varieties and curves
..................... 113
6.
Definiteness properties Over number fields
............... 120
7.
Non-degenerate heights and Euclidean spaces
.............. 124
8.
Mumford s theorem
......................... 134
CHAPTER
6
The Mordell-Weil Theorem
..................... 138
1. Kummer
theory
.......................... 139
2.
The weak Mordell-Weil theorem
................... 144
3.
The infinite descent
......................... 145
4.
Reduction steps
........................... 146
5.
Points of bounded height
...................... 149
6.
Theorem of the base
........................ 153
CHAPTER
7
The Thue-Siegel-Roth Theorem
.................. 158
1.
Statement of the theorem
...................... 158
2.
Reduction to simultaneous approximations
.............. 163
3.
Basic steps of the proof
....................... 165
4.
A combinatorial lemma
....................... 170
5.
Proof of Proposition
3.1...................... 171
6.
Wronskians
............................ 173
7.
Factorization of a polynomial
.................... 175
8.
The index
............................. ¡78
9.
Proof of Proposition
3.2...................... 181
10.
A geometric formulation of Roth s theorem
.............. 183
Contents xiii
CHAPTER
8
Siegel s Theorem and Integral Points
................ 188
1.
Height of integral points
....................... 189
2.
Finiteness theorems
......................... 192
3.
The curve ax
+
by
= 1....................... 194
4.
The Thue-Siegel curve
........................ 196
5.
Curves of genus
0.......................... 197
6.
Torsion points on curves
....................... 200
7.
Division points on curves
...................... 205
8.
Non-abelian
Kummer
theory
..................... 212
CHAPTER
9
Hubert s Irreducibility Theorem
................... 225
1.
Irreducibility and integral points
................... 226
2.
Irreducibility Over the rational numbers
................ 229
3.
Reduction steps
.......................... 233
4.
Function fields
........................... 236
5.
Abstract definition of Hubert sets
................... 239
6.
Applications to commutative group varieties
.............. 242
CHAPTER
10
Weil Functions and
Néron
Divisors
................. 247
1.
Bounded sets and functions
..................... 247
2.
Néron
divisors and Weil functions
.................. 252
3.
Positive divisors
.......................... 258
4.
The associated height function
.................... 263
CHAPTER
11
Néron
Functions on Abelian Varieties
................ 266
1.
Existence of
Néron
functions
..................... 266
2.
Translation properties of
Néron
functions
............... 271
3.
Néron
functions on varieties
..................... 276
4.
Reciprocity laws
.......................... 283
5.
Néron
functions as intersection multiplicities
.............. 286
6.
The
Néron
symbol and group extensions
................ 290
CHAPTER
12
Algebraic Families of
Néron
Functions
............... 296
1.
Variation of
Néron
functions in an algebraic family
........... 297
2.
Silverman s height and specialization theorems
............. 303
3.
Néron
heights as intersection multiplicities
............... 307
4.
Fibral divisors
........................... 314
5.
The height determined by a section
:
Tate s theorem
........... 320
xiv Contents
CHAPTER
13
Néron
Functions Over the Complex Numbers
............ 324
1.
The
Néron
function of an abelian variety
................ 324
2.
The scalar product of differentials of first kind
............. 327
3.
The canonical 2-form and the Riemann theta function
.......... 332
4.
The divisor of the Riemann theta function
............... 334
5.
Green,
Néron,
and theta functions
.................. 339
6.
The law of interchange of argument and parameter
........... 341
7.
Differentials of third kind and Green s function
............. 344
Appendix
..............................347
Review of S. Lang s Diophantine Geometry, by L. J. Mordell
.........349
Review of
L. J. Mordelľs
Diophantine Equations, by S. Lang
.........355
Bibliography
............................359
Index
................................367
|
| any_adam_object | 1 |
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| dewey-full | 512/.74 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.74 |
| dewey-search | 512/.74 |
| dewey-sort | 3512 274 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| isbn | 0387908374 3540908374 9783540908371 9780387908373 |
| language | English |
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| spelling | Lang, Serge 1927-2005 Verfasser (DE-588)119305119 aut Fundamentals of diophantine geometry Serge Lang New York [u.a.] Springer 1983 XVIII, 370 S. txt rdacontent n rdamedia nc rdacarrier Frühere Ausg. u.d.T.: Lang, Serge: Diophantine geometry Géométrie algébrique arithmétique Arithmetical algebraic geometry Diophantische Geometrie (DE-588)4150021-0 gnd rswk-swf Diophantische Gleichung (DE-588)4012386-8 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Diophantische Geometrie (DE-588)4150021-0 s DE-604 Diophantische Gleichung (DE-588)4012386-8 s 1\p DE-604 Früher u.d.T. Diophantine geometry Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000174042&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lang, Serge 1927-2005 Fundamentals of diophantine geometry Géométrie algébrique arithmétique Arithmetical algebraic geometry Diophantische Geometrie (DE-588)4150021-0 gnd Diophantische Gleichung (DE-588)4012386-8 gnd |
| subject_GND | (DE-588)4150021-0 (DE-588)4012386-8 (DE-588)4151278-9 |
| title | Fundamentals of diophantine geometry |
| title_auth | Fundamentals of diophantine geometry |
| title_exact_search | Fundamentals of diophantine geometry |
| title_full | Fundamentals of diophantine geometry Serge Lang |
| title_fullStr | Fundamentals of diophantine geometry Serge Lang |
| title_full_unstemmed | Fundamentals of diophantine geometry Serge Lang |
| title_old | Diophantine geometry |
| title_short | Fundamentals of diophantine geometry |
| title_sort | fundamentals of diophantine geometry |
| topic | Géométrie algébrique arithmétique Arithmetical algebraic geometry Diophantische Geometrie (DE-588)4150021-0 gnd Diophantische Gleichung (DE-588)4012386-8 gnd |
| topic_facet | Géométrie algébrique arithmétique Arithmetical algebraic geometry Diophantische Geometrie Diophantische Gleichung Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000174042&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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