The arithmetic of elliptic curves:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht
Springer
2009
|
| Edition: | 2. Edition |
| Series: | Graduate texts in mathematics
106 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XX, 513 Seiten Illustrationen 235 mm x 155 mm |
| ISBN: | 9780387094939 9780387094946 |
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MARC
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| 100 | 1 | |a Silverman, Joseph H. |d 1955- |e Verfasser |0 (DE-588)118906933 |4 aut | |
| 245 | 1 | 0 | |a The arithmetic of elliptic curves |c Joseph H. Silverman |
| 250 | |a 2. Edition | ||
| 264 | 1 | |a Dordrecht |b Springer |c 2009 | |
| 300 | |a XX, 513 Seiten |b Illustrationen |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 106 | |
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| 650 | 4 | |a Arithmetic | |
| 650 | 4 | |a Curves, Algebraic | |
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Record in the Search Index
| _version_ | 1806142889591308288 |
|---|---|
| adam_text |
Contents
Preface
to the Second Edition
v
Preface to the First Edition
vii
Introduction
xvii
CHAPTER I
Algebraic Varieties
1
§ 1.
Affine
Varieties
1
§2.
Projective Varieties
6
§3.
Maps Between Varieties
11
Exercises
14
CHAPTER II
Algebraic Curves
17
§1.
Curves
17
§2.
Maps Between Curves
19
§3.
Divisors
27
§4.
Differentials
30
§5.
The Riemann-Roch Theorem
33
Exercises
37
CHAPTER III
The Geometry of Elliptic Curves
41
§1.
Weierstrass
Equations
42
§2.
The Group Law
51
§3.
Elliptic Curves
58
§4. Isogenies 66
85.
The Invariant Differential
75
xi
xii
Contents
§6.
The Dual Isogeny
80
§7.
The
Tate
Module
87
§8.
The Weil Pairing
92
§9.
The Endomorphism Ring
99
§10.
The Automorphism Group
103
Exercises
104
CHAPTER IV
The Formal Group of an Elliptic Curve
115
§ 1.
Expansion Around
О
115
§2.
Formal Groups
120
§3.
Groups Associated to Formal Groups
123
§4.
The Invariant Differential
125
§5.
The Formal Logarithm
127
§6.
Formal Groups over Discrete Valuation Rings
129
§7.
Formal Groups in Characteristic
ρ
132
Exercises
135
CHAPTER V
Elliptic Curves over Finite Fields
137
§ 1.
Number of Rational Points
137
§2.
The Weil Conjectures
140
§3.
The Endomorphism Ring
144
§4.
Calculating the
Hasse
Invariant
148
Exercises
153
CHAPTER VI
Elliptic Curves over
С
157
§1.
Elliptic Integrals
158
§2.
Elliptic Functions
161
§3.
Construction of Elliptic Functions
165
§4.
Maps Analytic and Maps Algebraic
171
§5.
Uniformization
173
§6.
The Lefschetz Principle
177
Exercises
178
Contents xiii
CHAPTER
VII
Elliptic Curves over Local Fields
185
§1.
Minimal
Weierstrass
Equations
185
§2.
Reduction Modulo
π
187
§3.
Points of Finite Order
192
§4.
The Action of Inertia
194
§5.
Good and Bad Reduction
196
§6.
The Group E/Eo
199
§7.
The Criterion of
Néron-Ogg-Shafarevich
201
Exercises
203
CHAPTER
VIII
Elliptic Curves over Global Fields
207
§1.
The Weak Mordell-Weil Theorem
208
§2.
The
Kummer
Pairing via Cohomology
215
§3.
The Descent Procedure
218
§4.
The Mordell-Weil Theorem over
Q
220
§5.
Heights on
Projective
Space
224
§6.
Heights on Elliptic Curves
234
§7.
Torsion Points
240
§8.
The Minimal Discriminant
243
§9.
The Canonical Height
247
§10.
The Rank of an Elliptic Curve
254
§11.
Szpiro's Conjecture and ABC
255
Exercises
261
CHAPTER IX
Integral Points on Elliptic Curves
269
§1.
Diophantine Approximation
270
§2.
Distance Functions
273
§3.
Siegel's Theorem
276
§4.
The S-Unit Equation
281
§5.
Effective Methods
286
§6.
Shafarevich's Theorem
293
§7.
The Curve Y2=X3 + D
296
§8.
Roth's Theorem
—
An Overview
299
Exercises
302
xiv Contents
CHAPTER X
Computing the Mordell-Weil Group
309
§ 1.
An Example
310
§2.
Twisting
—
General Theory
318
§3.
Homogeneous Spaces
321
§4.
The
Selmer
and Shafarevich-Tate Groups
331
§5.
Twisting
—
Elliptic Curves
341
§6.
The Curve Y2
=
X3
+
DX
344
Exercises
355
CHAPTER XI
Algorithmic Aspects of Elliptic Curves
363
§1.
Double-and-Add Algorithms
364
§2.
Lenstra's Elliptic Curve Factorization Algorithm
366
§3.
Counting the Number of Points in E(¥g)
372
§4.
Elliptic Curve Cryptography
376
§5.
Solving the ECDLP: The General Case
381
§6.
Solving the ECDLP: Special Cases
386
§7.
Pairing-Based Cryptography
390
§8.
Computing the Weil Pairing
393
§9.
The Tate-Lichtenbaum Pairing
397
Exercises
403
APPENDIX A
Elliptic Curves in Characteristics
2
and
3 409
Exercises
414
APPENDIX
В
Group Cohomology
(Я0
and
Я1)
415
§1.
Cohomology of Finite Groups
415
§2.
Galois Cohomology
418
§3.
Nonabelian Cohomology
421
Exercises
422
Contents xv
APPENDIX
С
Further Topics: An Overview
425
§11.
Complex Multiplication
425
§12.
Modular Functions
429
§13.
Modular Curves
439
§14.
Tate
Curves
443
§15.
Néron
Models and Tate's Algorithm
446
§16.
L-Series
449
§17.
Duality Theory
453
§18.
Local Height Functions
454
§19.
The Image of Galois
455
§20.
Function Fields and Specialization Theorems
456
§21.
Variation of ap and the Sato-Tate Conjecture
458
Notes on Exercises
461
List of Notation
467
References
473
Index
489 |
| any_adam_object | 1 |
| author | Silverman, Joseph H. 1955- |
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| classification_tum | MAT 145f |
| ctrlnum | (OCoLC)1411781934 (DE-599)DNB992999677 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/5 |
| dewey-search | 516.3/5 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. Edition |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-08-01T00:28:05Z |
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| isbn | 9780387094939 9780387094946 |
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| publishDate | 2009 |
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| publisher | Springer |
| record_format | marc |
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| series2 | Graduate texts in mathematics |
| spelling | Silverman, Joseph H. 1955- Verfasser (DE-588)118906933 aut The arithmetic of elliptic curves Joseph H. Silverman 2. Edition Dordrecht Springer 2009 XX, 513 Seiten Illustrationen 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 106 Algebraïsche krommen gtt Arithmetic Curves, Algebraic Curves, Elliptic Elliptische Kurve (DE-588)4014487-2 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Algebraische Varietät (DE-588)4581715-7 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Diophantische Approximation (DE-588)4135760-7 gnd rswk-swf Elliptische Kurve (DE-588)4014487-2 s Algebra (DE-588)4001156-2 s Algebraische Zahlentheorie (DE-588)4001170-7 s DE-604 Diophantische Approximation (DE-588)4135760-7 s 1\p DE-604 Algebraische Varietät (DE-588)4581715-7 s 2\p DE-604 Graduate texts in mathematics 106 (DE-604)BV000000067 106 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3253905&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017705554&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Silverman, Joseph H. 1955- The arithmetic of elliptic curves Graduate texts in mathematics Algebraïsche krommen gtt Arithmetic Curves, Algebraic Curves, Elliptic Elliptische Kurve (DE-588)4014487-2 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Algebraische Varietät (DE-588)4581715-7 gnd Algebra (DE-588)4001156-2 gnd Diophantische Approximation (DE-588)4135760-7 gnd |
| subject_GND | (DE-588)4014487-2 (DE-588)4001170-7 (DE-588)4581715-7 (DE-588)4001156-2 (DE-588)4135760-7 |
| title | The arithmetic of elliptic curves |
| title_auth | The arithmetic of elliptic curves |
| title_exact_search | The arithmetic of elliptic curves |
| title_full | The arithmetic of elliptic curves Joseph H. Silverman |
| title_fullStr | The arithmetic of elliptic curves Joseph H. Silverman |
| title_full_unstemmed | The arithmetic of elliptic curves Joseph H. Silverman |
| title_short | The arithmetic of elliptic curves |
| title_sort | the arithmetic of elliptic curves |
| topic | Algebraïsche krommen gtt Arithmetic Curves, Algebraic Curves, Elliptic Elliptische Kurve (DE-588)4014487-2 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Algebraische Varietät (DE-588)4581715-7 gnd Algebra (DE-588)4001156-2 gnd Diophantische Approximation (DE-588)4135760-7 gnd |
| topic_facet | Algebraïsche krommen Arithmetic Curves, Algebraic Curves, Elliptic Elliptische Kurve Algebraische Zahlentheorie Algebraische Varietät Algebra Diophantische Approximation |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3253905&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017705554&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT silvermanjosephh thearithmeticofellipticcurves |