Advanced topics in the arithmetic of elliptic curves:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1999
|
| Ausgabe: | Corr. 2. print |
| Schriftenreihe: | Graduate texts in mathematics
151 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 525 S. graph. Darst. |
| ISBN: | 3540943285 3540943250 0387943285 0387943250 |
Internformat
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 Advanced topics in the arithmetic of elliptic curves |c Joseph H. Silverman |
| 250 | |a Corr. 2. print | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1999 | |
| 300 | |a XIII, 525 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 151 | |
| 650 | 4 | |a Arithmetic | |
| 650 | 4 | |a Curves, Algebraic | |
| 650 | 4 | |a Curves, Elliptic | |
| 650 | 0 | 7 | |a Elliptische Kurve |0 (DE-588)4014487-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Arithmetik |0 (DE-588)4002919-0 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Arithmetik |0 (DE-588)4002919-0 |D s |
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| 830 | 0 | |a Graduate texts in mathematics |v 151 |w (DE-604)BV000000067 |9 151 | |
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Datensatz im Suchindex
| _version_ | 1804127569079959552 |
|---|---|
| adam_text | Contents
Preface
vii
Computer
Packages
viii
Acknowledgments
viii
Introduction
1
CHAPTER I
Elliptic and Modular Functions
5
§1.
The Modular Group
6
§2.
The Modular Curve
λ (1)
14
§3.
Modular Functions
23
§4.
Uniformization and Fields of Moduli
34
§5.
Elliptic Functions Revisited
38
§6.
q-Expansions of Elliptic Functions
47
§7.
g-Expansions of Modular Functions
55
§8.
Jacobi s Product Formula for
Δ(τ)
62
§9. Hecke
Operators
67
§10. Hecke
Operators Acting on Modular Forms
74
§11.
¿-Series Attached to Modular Forms
80
Exercises
85
CHAPTER II
Complex Multiplication
95
§1.
Complex Multiplication over
С
96
§2.
Rationality Questions
104
§3.
Class Field Theory
—
A Brief Review
115
§4.
The Hubert Class Field
121
§5.
The Maximal Abelian Extension
128
§6,
Integrality of
j
140
§7.
Cyclotomic Class Field Theory
151
§8.
The
Maia
Theorem of Complex Multiplication
157
§9.
The Associated Grossenenaracter
165
§10.
The i-Series Attached to a CM Elliptic Curve
171
Exercises
178
xii Contents
CHAPTER
III
Elliptic Surfaces
187
§1.
Elliptic Curves over Function Fields
188
§2.
The Weak Mordell-Weil Theorem
191
§3.
Elliptic Surfaces
200
§4.
Heights on Elliptic Curves over Function Fields
212
§5.
Split Elliptic Surfaces and Sets of Bounded Height
220
§6.
The Mordell-Weil Theorem for Function Fields
230
§7.
The Geometry of Algebraic Surfaces
231
§8.
The Geometry of Fibered Surfaces
236
§9.
The Geometry of Elliptic Surfaces
245
§10.
Heights and Divisors on Varieties
255
§11.
Specialization Theorems for Elliptic Surfaces
265
§12.
Integral Points on Elliptic Curves over Function Fields
274
Exercises
278
CHAPTER IV
The
Néron
Model
289
§1.
Group Varieties
290
§2.
Schemes and S-Schemes
297
§3.
Group Schemes
306
§4.
Arithmetic Surfaces
311
§5.
Néron
Models
318
§6.
Existence of
Néron
Models
325
§7.
Intersection Theory, Minimal Models, and Blowing-Up
338
§8.
The Special Fiber of
a Néron
Model
350
§9.
Tate s Algorithm to Compute the Special Fiber
361
§10.
The Conductor of an Elliptic Curve
379
§11.
Ogg s Formula
389
Exercises
396
CHAPTER V
Elliptic Curves over Complete Fields
408
§1.
Elliptic Curves over
С
408
§2.
Elliptic Curves over
R
413
§3.
The
Tate
Curve
422
§4.
The
Tate
Map Is Surjective
429
§5.
Elliptic Curves over p-adic Fields
438
§6.
Some Applications of p-adic Uniformization
445
Exercises
448
Contents xiii
CHAPTER VI
Local Height Functions
454
§1.
Existence of Local Height Functions
455
§2.
Local Decomposition of the Canonical Height
461
§3.
Archimedean Absolute Values
—
Explicit Formulas
463
§4.
Non-Archimedean Absolute Values
—
Explicit Formulas
469
Exercises
476
APPENDIX A
Some Useful Tables
481
§1.
Bernoulli Numbers and
ζ(2&)
481
§2.
Fourier Coefficients of
Δ(τ)
and
j
(τ)
482
§3.
Elliptic Curves over
Q
with Complex Multiplication
483
Notes on Exercises
484
References
488
List of Notation
498
Index
504
|
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| author | Silverman, Joseph H. 1955- |
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| dewey-full | 516.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/52 |
| dewey-search | 516.3/52 |
| dewey-sort | 3516.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Corr. 2. print |
| format | Book |
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| id | DE-604.BV012882778 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:35:26Z |
| institution | BVB |
| isbn | 3540943285 3540943250 0387943285 0387943250 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008769344 |
| oclc_num | 50573659 |
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| owner | DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-11 |
| physical | XIII, 525 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Silverman, Joseph H. 1955- Verfasser (DE-588)118906933 aut Advanced topics in the arithmetic of elliptic curves Joseph H. Silverman Corr. 2. print New York [u.a.] Springer 1999 XIII, 525 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 151 Arithmetic Curves, Algebraic Curves, Elliptic Elliptische Kurve (DE-588)4014487-2 gnd rswk-swf Arithmetik (DE-588)4002919-0 gnd rswk-swf Elliptische Kurve (DE-588)4014487-2 s Arithmetik (DE-588)4002919-0 s DE-604 Graduate texts in mathematics 151 (DE-604)BV000000067 151 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008769344&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Silverman, Joseph H. 1955- Advanced topics in the arithmetic of elliptic curves Graduate texts in mathematics Arithmetic Curves, Algebraic Curves, Elliptic Elliptische Kurve (DE-588)4014487-2 gnd Arithmetik (DE-588)4002919-0 gnd |
| subject_GND | (DE-588)4014487-2 (DE-588)4002919-0 |
| title | Advanced topics in the arithmetic of elliptic curves |
| title_auth | Advanced topics in the arithmetic of elliptic curves |
| title_exact_search | Advanced topics in the arithmetic of elliptic curves |
| title_full | Advanced topics in the arithmetic of elliptic curves Joseph H. Silverman |
| title_fullStr | Advanced topics in the arithmetic of elliptic curves Joseph H. Silverman |
| title_full_unstemmed | Advanced topics in the arithmetic of elliptic curves Joseph H. Silverman |
| title_short | Advanced topics in the arithmetic of elliptic curves |
| title_sort | advanced topics in the arithmetic of elliptic curves |
| topic | Arithmetic Curves, Algebraic Curves, Elliptic Elliptische Kurve (DE-588)4014487-2 gnd Arithmetik (DE-588)4002919-0 gnd |
| topic_facet | Arithmetic Curves, Algebraic Curves, Elliptic Elliptische Kurve Arithmetik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008769344&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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