Umfang:
Online-Ressource
,
v.: digital
Ausgabe:
Online-Ausg. Springer eBook Collection. Mathematics and Statistics Electronic reproduction; Available via World Wide Web
ISBN:
1282827219
,
9783034601283
,
9781282827219
Serie:
Progress in Mathematics 277
Inhalt:
This book examines the range of available tools from analytic number theory that can be applied to study the density of rational points on projective varieties.
Inhalt:
This monograph is concerned with counting rational points of bounded height on projective algebraic varieties. This is a relatively young topic, whose exploration has already uncovered a rich seam of mathematics situated at the interface of analytic number theory and Diophantine geometry. The goal of the book is to give a systematic account of the field with an emphasis on the role played by analytic number theory in its development. Among the themes discussed in detail are the Manin conjecture for del Pezzo surfaces, Heath-Brown's dimension growth conjecture, and the Hardy-Littlewood circle method. Readers of this monograph will be rapidly brought into contact with a spectrum of problems and conjectures that are central to this fertile subject area.
Anmerkung:
"This book is based on a short graduate course given by the author at the I.C.T.P. School and Conference on Analytic Number Theory, during the period 23rd April to 11th May, 2007."--Preface
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Includes bibliographical references and index
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CONTENTS; Preface; 1 Introduction; 1.1 A naive heuristic; 1.2 The basic counting function; 1.3 Influence of analytic number theory; Exercises for Chapter 1; 2 The Manin conjectures; 2.1 Divisors on varieties; 2.2 The conjectures; 2.3 Degree 3; 2.4 Degree 4; 2.5 Degree ≥ 5; 2.6 Universal torsors; Exercises for Chapter 2; 3 The dimension growth conjecture; 3.1 Linear spaces on hypersurfaces; 3.2 Dimension growth for hypersurfaces; 3.3 Exponential sums; 3.4 Covering with linear spaces; Exercises for Chapter 3; 4 Uniform bounds for curves and surfaces; 4.1 The determinant method
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4.2 The geometry of numbers4.3 General plane curves; 4.4 Diagonal plane curves; Exercises for Chapter 4; 5 A 1 del Pezzo surface of degree 6; 5.1 Passage to the universal torsor; 5.2 The asymptotic formula; 5.3 Perron's formula; Exercises for Chapter 5; 6 D 4 del Pezzo surface of degree 3; 6.1 Passage to the universal torsor; 6.2 A crude upper bound; 6.3 A better upper bound; Exercises for Chapter 6; 7 Siegel's lemma and non-singular surfaces; 7.1 Dual variety; 7.2 Non-singular del Pezzo surfaces of degree 3; 7.3 Non-singular del Pezzo surfaces of degree 4; Exercises for Chapter 7
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8 The Hardy-Littlewood circle method8.1 Major arcs and minor arcs; 8.2 Quartic hypersurfaces; 8.3 Diagonal cubic surfaces; Exercises for Chapter 8; Bibliography; Index
,
Electronic reproduction; Available via World Wide Web
Weitere Ausg.:
ISBN 9783034601290
Weitere Ausg.:
ISBN 9783034601283
Weitere Ausg.:
ISBN 1282827197
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe Browning, Tim, 1976 - Quantitative arithmetic of projective varieties Basel : Birkhäuser, 2009 ISBN 9783034601283
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Projektive Varietät
;
Diophantische Geometrie
;
Analytische Zahlentheorie
DOI:
10.1007/978-3-0346-0129-0
URL:
Volltext
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