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Point-counting and the Zilber-Pink conjecture

Record Type:	Electronic resources : Monograph/item
Title/Author:	Point-counting and the Zilber-Pink conjectureJonathan Pila.
Author:	Pila, Jonathan.
Published:	Cambridge :Cambridge University Press,2022.
Description:	x, 254 p. :ill., digital ;23 cm.
Notes:	Title from publisher's bibliographic system (viewed on 07 Apr 2022).
Subject:	Arithmetical algebraic geometry.
Online resource:	https://doi.org/10.1017/9781009170314
ISBN:	9781009170314$q(electronic bk.)

Point-counting and the Zilber-Pink conjecture
Pila, Jonathan.

Point-counting and the Zilber-Pink conjecture[electronic resource] /Jonathan Pila. - Cambridge :Cambridge University Press,2022. - x, 254 p. :ill., digital ;23 cm. - Cambridge tracts in mathematics ;228. - Cambridge tracts in mathematics ;96..

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.

ISBN: 9781009170314$q(electronic bk.)Subjects--Topical Terms:

347677
Arithmetical algebraic geometry.

LC Class. No.: QA242.5 / .P553 2022

Dewey Class. No.: 516.35

Point-counting and the Zilber-Pink conjecture
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490 1 $a Cambridge tracts in mathematics ; $v 228
500 $a Title from publisher's bibliographic system (viewed on 07 Apr 2022).
505 0 $a 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.
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.
650 0 $a Arithmetical algebraic geometry. $3 347677
650 0 $a Diophantine equations. $3 387265
650 0 $a Modular curves. $3 560023
650 0 $a Model theory. $3 191661
830 0 $a Cambridge tracts in mathematics ; $v 96. $3 829132
856 4 0 $u https://doi.org/10.1017/9781009170314