Quantitative Arithmetic of Projective Varieties

OverthemillenniaDiophantineequationshavesuppliedanextremelyfertilesource ofproblems. Their study hasilluminated everincreasingpoints ofcontactbetween very di?erent subject areas, including algebraic geometry, mathematical logic, - godictheoryandanalyticnumber theory. Thefocus ofthis bookisonthe interface of algebraic geometry with analytic number theory, with the basic aim being to highlight the ro ^le that analytic number theory has to play in the study of D- phantine equations. Broadly speaking, analytic number theory can be characterised as a subject concerned with counting interesting objects. Thus, in the setting of Diophantine geometry, analytic number theory is especially suited to questions concerning the "distribution" of integral and rational points on algebraic varieties. Determining the arithmetic of a?ne varieties, both qualitatively and quantitatively, is much more complicated than for projective varieties. Given the breadth of the domain and the inherent di?culties involved, this book is therefore dedicated to an exp- ration of the projective setting. 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. It is a pleasure to thank Professors Balasubra- nian, Deshouillers and Kowalski for organising this meeting. Thanks are also due to Michael Harvey and Daniel Loughran for spotting several typographical errors in an earlier draft of this book. Over the years, the author has greatly bene?ted fromdiscussing mathematicswithProfessorsde la Bret` eche,Colliot-Th¿ el` ene,F- vry, Hooley, Salberger, Swinnerton-Dyer and Wooley.

"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. TOC:Preface.- 1. Introduction.- 2. The Manin Conjectures.- 3. The Dimension Growth Conjecture.- 4. Uniform Bounds for Curves and Surfaces.- 5. A1 Del Pezzo Surface of Degree 6.- 6. D4 Del Pezzo Surface of Degree 3.- 7. Siegel's Lemma and Non-singular Surfaces.- 8. The Hardy-Littlewood Circle Method.- Bibliography.- Index."

The Manin conjectures.- The dimension growth conjecture.- Uniform bounds for curves and surfaces.- A1 del Pezzo surface of degree 6.- D4 del Pezzo surface of degree 3.- Siegel's lemma and non-singular surfaces.- The Hardy-Littlewood circle method.