國立虎尾科技大學 |

Point-counting and the Zilber-Pink conjecture /

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Point-counting and the Zilber-Pink conjecture // Jonathan Pila.
Author:	Pila, Jonathan,
Description:	1 online resource (x, 254 pages) :digital, PDF file(s). :
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 (ebook)

Point-counting and the Zilber-Pink conjecture /
Pila, Jonathan,1962-

Point-counting and the Zilber-Pink conjecture /Jonathan Pila. - 1 online resource (x, 254 pages) :digital, PDF file(s). - Cambridge tracts in mathematics ;228. - Cambridge tracts in mathematics ;203..

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 (ebook)Subjects--Topical Terms:

680690
Arithmetical algebraic geometry.

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

Dewey Class. No.: 516.3/5

Point-counting and the Zilber-Pink conjecture /
LDR:02822nam a2200301 i 4500 001 1126306
003 UkCbUP
005 20220609050103.0
006 m|||||o||d||||||||
007 cr||||||||||||
008 240926s2022||||enk o ||1 0|eng|d
020 $a 9781009170314 (ebook)
020 $z 9781009170321 (hardback)
035 $a CR9781009170314
040 $a UkCbUP $b eng $e rda $c UkCbUP
050 0 0 $a QA242.5 $b .P553 2022
082 0 0 $a 516.3/5 $2 23/eng20220215
100 1 $a Pila, Jonathan, $d 1962- $e author. $3 1445055
245 1 0 $a Point-counting and the Zilber-Pink conjecture / $c Jonathan Pila.
264 1 $a Cambridge : $b Cambridge University Press, $c 2022.
300 $a 1 online resource (x, 254 pages) : $b digital, PDF file(s).
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
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 680690
650 0 $a Diophantine equations. $3 792270
650 0 $a Modular curves. $3 890182
650 0 $a Model theory. $3 633609
776 0 8 $i Print version: $z 9781009170321
830 0 $a Cambridge tracts in mathematics ; $v 203. $3 1238301
856 4 0 $u https://doi.org/10.1017/9781009170314