Point-counting and the Zilber-Pink conjecture:
Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the André-Oort and Zilber-Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that a...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, VIC, Australia ; New Delhi, India ; Singapore
Cambridge University Press
2022
|
| Schriftenreihe: | Cambridge tracts in mathematics
228 |
| Schlagworte: | |
| Online-Zugang: | BSB01 BTU01 FHN01 TUM01 TUM02 Volltext |
| Zusammenfassung: | Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the André-Oort and Zilber-Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only a basic familiarity with number theory and complex algebraic geometry. It serves as an introduction for postgraduate students and researchers to the main ideas, results, problems, and themes of current research |
| Beschreibung: | Title from publisher's bibliographic system (viewed on 07 Apr 2022) Point-counting -- Multiplicative Manin-Mumford -- Powers of the modular curve as Shimura varieties -- Modular André-Oort -- Point-counting and the André-Oort conjecture -- Model theory and definable sets -- O-minimal structures -- Parameterization and point-counting -- Better bounds -- Point-counting and Galois orbit bounds -- Complex analysis in O-minimal structures -- Schanuel's conjecture and Ax-Schanuel -- A formal setting -- Modular Ax-Schanuel -- Ax-Schanuel for Shimura varieties -- Quasi-periods of elliptic curves -- Sources -- Formulations -- Some results -- Curves in a power of the modular curve -- Conditional modular Zilber-Pink -- O-minimal uniformity -- Uniform Zilber-Pink |
| Beschreibung: | 1 Online-Ressource (x, 254 Seiten) Illustrationen |
| ISBN: | 9781009170314 |
| DOI: | 10.1017/9781009170314 |
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| 520 | |a Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the André-Oort and Zilber-Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only a basic familiarity with number theory and complex algebraic geometry. It serves as an introduction for postgraduate students and researchers to the main ideas, results, problems, and themes of current research | ||
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Datensatz im Suchindex
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| discipline | Mathematik |
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| isbn | 9781009170314 |
| language | English |
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| spelling | Pila, Jonathan 1962- (DE-588)1261762185 aut Point-counting and the Zilber-Pink conjecture Jonathan Pila Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, VIC, Australia ; New Delhi, India ; Singapore Cambridge University Press 2022 1 Online-Ressource (x, 254 Seiten) Illustrationen txt rdacontent c rdamedia cr rdacarrier Cambridge tracts in mathematics 228 Title from publisher's bibliographic system (viewed on 07 Apr 2022) Point-counting -- Multiplicative Manin-Mumford -- Powers of the modular curve as Shimura varieties -- Modular André-Oort -- Point-counting and the André-Oort conjecture -- Model theory and definable sets -- O-minimal structures -- Parameterization and point-counting -- Better bounds -- Point-counting and Galois orbit bounds -- Complex analysis in O-minimal structures -- Schanuel's conjecture and Ax-Schanuel -- A formal setting -- Modular Ax-Schanuel -- Ax-Schanuel for Shimura varieties -- Quasi-periods of elliptic curves -- Sources -- Formulations -- Some results -- Curves in a power of the modular curve -- Conditional modular Zilber-Pink -- O-minimal uniformity -- Uniform Zilber-Pink Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the André-Oort and Zilber-Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only a basic familiarity with number theory and complex algebraic geometry. It serves as an introduction for postgraduate students and researchers to the main ideas, results, problems, and themes of current research Arithmetical algebraic geometry Diophantine equations Modular curves Model theory (DE-588)1071861417 Konferenzschrift gnd-content Erscheint auch als Druck-Ausgabe 978-1-00-917032-1 Cambridge tracts in mathematics 228 (DE-604)BV047362617 228 https://doi.org/10.1017/9781009170314 Verlag URL des Erstveröffentlichers Volltext |
| spellingShingle | Pila, Jonathan 1962- Point-counting and the Zilber-Pink conjecture Cambridge tracts in mathematics Arithmetical algebraic geometry Diophantine equations Modular curves Model theory |
| subject_GND | (DE-588)1071861417 |
| title | Point-counting and the Zilber-Pink conjecture |
| title_auth | Point-counting and the Zilber-Pink conjecture |
| title_exact_search | Point-counting and the Zilber-Pink conjecture |
| title_exact_search_txtP | Point-counting and the Zilber-Pink conjecture |
| title_full | Point-counting and the Zilber-Pink conjecture Jonathan Pila |
| title_fullStr | Point-counting and the Zilber-Pink conjecture Jonathan Pila |
| title_full_unstemmed | Point-counting and the Zilber-Pink conjecture Jonathan Pila |
| title_short | Point-counting and the Zilber-Pink conjecture |
| title_sort | point counting and the zilber pink conjecture |
| topic | Arithmetical algebraic geometry Diophantine equations Modular curves Model theory |
| topic_facet | Arithmetical algebraic geometry Diophantine equations Modular curves Model theory Konferenzschrift |
| url | https://doi.org/10.1017/9781009170314 |
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