Some problems of unlikely intersections in arithmetic and geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton ; Oxford
Princeton University Press
2012
|
| Series: | Annals of mathematics studies
181 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xi, 160 Seiten |
| ISBN: | 9780691153704 9780691153711 |
Staff View
MARC
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| 245 | 1 | 0 | |a Some problems of unlikely intersections in arithmetic and geometry |c Umberto Zannier ; with appendixes by David Masser |
| 264 | 1 | |a Princeton ; Oxford |b Princeton University Press |c 2012 | |
| 264 | 4 | |c © 2012 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
preface
ix
Notation and Conventions xi
Introduction: An Overview of Some Problems of Unlikely
Intersections
1
1
Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture
15
1.1
Torsion points on subvarieties of G^
.......................... 16
1.2
Higher multiplicative rank
................................ 22
1.3
Remarks on Theorem
1.3
and its developments
.................... 29
1.3.1
Fields other than
Q
................................ 29
1.3.2
Weakened assumptions
.............................. 29
1.3.3
Unlikely intersections of positive dimension and height bounds
........ 31
1.3.4
Unlikely intersections of positive dimension and Zilber s
........
conjecture
..................................... 33
1.3.5
Unlikely intersections and reducibility of lacunary poly-
........
nomials (Schinzel s conjecture)
......................... 35
1.3.6
Zhang s notion of dependence
.......................... 36
1.3.7
Abelian varieties (and other algebraic groups)
................. 36
1.3.8
Uniformity of bounds
............................... 37
Notes to Chapter
1....................................... 39
Sparseness of multiplicatively dependent points
.................... 39
Other unlikely intersections
............................... 39
A generalization of Theorem
1.3 ............................ 40
An application of the methods to zeros of linear recurrences
............. 40
Comments on the Methods
................................ 41
2
An Arithmetical Analogue
43
2.1
Some unlikely intersections in number fields
...................... 43
2.2
Some applications of Theorem
2.1 ........................... 48
2.3
An analogue of Theorem
2.1
for function fields
.................... 50
2.4
Some applications of Theorem
2.2 ........................... 52
2.5
A proof of Theorem
2.2................................. 54
Notes to Chapter
2....................................... 58
Simplifying the proof of Theorem
1.3.......................... 58
Rational points on curves over Wv
............................ 58
Unlikely Intersections and Holomorphic GCD in Nevanlinna Theory
........ 60
vi
CONTENTS
3
Unlikely Intersections in Elliptic Surfaces and Problems of
Masser 62
3.1
A method for the Manin-Mumford conjecture
..................... 62
3.2
Masser s questions on elliptic pencils
.......................... 66
3.3
A finiteness proof
.....................................
TO
3.4
Related problems, conjectures, and developments
................... 77
3.4.1
Pink s and related conjectures
.......................... 77
3.4.2
Extending Theorem
3.3
from
Q
to
С
...................... 80
3.4.3
Effectivity
..................................... 83
3.4.4
Extending Theorem
3.3
to arbitrary pairs of points on families
.......
of elliptic curves
................................. 84
3.4.5
Simple abelian surfaces and Pell s equations over function fields
....... 85
3.4.6
Further extensions and analogues
........................ 87
3.4.7
Dynamical analogues
............................... 89
Notes to Chapter
3....................................... 92
Torsion values for a single point: other arguments
.................. 92
A variation on the Manin-Mumford conjecture
.................... 93
Comments on the Methods
............................... 94
4
About the
André-Oort
Conjecture
96
4.1
Generalities about the
André-Oort
Conjecture
..................... 96
4.2
Modular curves and complex multiplication
...................... 99
4.3
The theorem of
André
.................................. 105
4.3.1
An effective variation
...............................
Ill
4.4
Pila s proof of Andre s theorem
............................. 112
4.5
Shimura varieties
..................................... 118
Notes to Chapter
4....................................... 123
Remarks on Edixhoven s approach to Andre s theorem
................ 123
Some unlikely intersections beyond
André-Oort
.................... 124
Definability and o-minimal structures
......................... 125
Appendix A Distribution of Rational Points on Subanalytic
Surfaces
128
by
Umberto Zannier
Appendix
В
Uniformity in Unlikely Intersections: An Example for Lines in
Three Dimensions
136
by David
Masser
Appendix
С
Silverman s Bounded Height Theorem for Elliptic Curves:
A Direct Proof
138
by David
Masser
Appendix
D
Lower Bounds for Degrees of Torsion Points: The Transcendence
Approach
140
by David
Masser
Appendix
E A
Transcendence Measure for a Quotient of Periods
143
bv David
Masser
CONTENTS
vii
Appendix F Counting Rational Points on Analytic Curves:
A Transcendence Approach
145
by David
Masser
Appendix
G
Mixed Problems: Another Approach
147
by David
Masser
Bibliography
149
Index
159
|
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| callnumber-first | Q - Science |
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| dewey-full | 516.3/5 |
| 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 |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:15:50Z |
| institution | BVB |
| isbn | 9780691153704 9780691153711 |
| language | English |
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| physical | xi, 160 Seiten |
| publishDate | 2012 |
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| publisher | Princeton University Press |
| record_format | marc |
| series | Annals of mathematics studies |
| series2 | Annals of mathematics studies |
| spelling | Zannier, Umberto 1957- Verfasser (DE-588)102371471X aut Some problems of unlikely intersections in arithmetic and geometry Umberto Zannier ; with appendixes by David Masser Princeton ; Oxford Princeton University Press 2012 © 2012 xi, 160 Seiten txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies number 181 Intersection theory Algebraic varieties Algebraic geometry Arithmetische Geometrie (DE-588)4131383-5 gnd rswk-swf Schnitttheorie (DE-588)4179890-9 gnd rswk-swf Schnitttheorie (DE-588)4179890-9 s Arithmetische Geometrie (DE-588)4131383-5 s DE-604 Masser, David William 1948- Sonstige (DE-588)102371499X oth Erscheint auch als Online-Ausgabe 9781400842711 Annals of mathematics studies 181 (DE-604)BV000000991 181 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024852306&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Zannier, Umberto 1957- Some problems of unlikely intersections in arithmetic and geometry Annals of mathematics studies Intersection theory Algebraic varieties Algebraic geometry Arithmetische Geometrie (DE-588)4131383-5 gnd Schnitttheorie (DE-588)4179890-9 gnd |
| subject_GND | (DE-588)4131383-5 (DE-588)4179890-9 |
| title | Some problems of unlikely intersections in arithmetic and geometry |
| title_auth | Some problems of unlikely intersections in arithmetic and geometry |
| title_exact_search | Some problems of unlikely intersections in arithmetic and geometry |
| title_full | Some problems of unlikely intersections in arithmetic and geometry Umberto Zannier ; with appendixes by David Masser |
| title_fullStr | Some problems of unlikely intersections in arithmetic and geometry Umberto Zannier ; with appendixes by David Masser |
| title_full_unstemmed | Some problems of unlikely intersections in arithmetic and geometry Umberto Zannier ; with appendixes by David Masser |
| title_short | Some problems of unlikely intersections in arithmetic and geometry |
| title_sort | some problems of unlikely intersections in arithmetic and geometry |
| topic | Intersection theory Algebraic varieties Algebraic geometry Arithmetische Geometrie (DE-588)4131383-5 gnd Schnitttheorie (DE-588)4179890-9 gnd |
| topic_facet | Intersection theory Algebraic varieties Algebraic geometry Arithmetische Geometrie Schnitttheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024852306&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000991 |
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