Point-counting and the Zilber-Pink conjecture / Jonathan Pila.
| Author/creator | Pila, Jonathan, 1962- author. |
| Format | Electronic |
| Publication | Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2022. |
| Copyright Date | ©2022 |
| Description | 1 online resource (x, 254 pages). |
| Supplemental Content | Cambridge University Press |
| Subjects |
| Series | Cambridge tracts in mathematics ; 228 Cambridge tracts in mathematics ; 228. ^A470109 |
| Contents | 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. |
| Abstract | Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the Andř-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. |
| Bibliography note | Includes bibliographical references and indexes. |
| Source of description | Description based on online resource; title from digital title page (viewed on June 03, 2022). |
| Issued in other form | Print version: Pila, Jonathan, 1962- Point-counting and the Zilber-Pink conjecture Cambridge ; New York, NY : Cambridge University Press, 2022 9781009170321 |
| LCCN | 2021060969 |
| ISBN | 9781009170314 electronic book |
| ISBN | 1009170317 electronic book |
| ISBN | hardcover |