Quantitative Arithmetic of Projective Varieties by Timothy D. BrowningQuantitative Arithmetic of Projective Varieties by Timothy D. Browning

Quantitative Arithmetic of Projective Varieties

byTimothy D. Browning

Hardcover | September 18, 2009

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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.
Title:Quantitative Arithmetic of Projective VarietiesFormat:HardcoverDimensions:160 pages, 23.5 × 15.5 × 0.01 inPublished:September 18, 2009Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:303460128X

ISBN - 13:9783034601283


Table of Contents

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.

Editorial Reviews

From the reviews:"The book under review considers the distribution of integral or rational points of bounded height on (projective) algebraic varieties. . well-written and well-organized. . Introductory material is discussed when appropriate, motivation and context are provided when necessary, and there are even small sets of exercises at the end of every chapter, making the book suitable for self or guided study . ." (Felipe Zaldivar, The Mathematical Association of America, January, 2010)"The most important feature of the book is the way it presents the geometric and analytic aspects of the theory on a unified equal footing. The interface between these two fields has been a very productive subject in recent years, and this book is likely to be of considerable value to anyone, graduate student and up, interested in this area." (Roger Heath-Brown, Zentralblatt MATH, Vol. 1188, 2010)"The book . is focused on exposing how tools rooted in analytic number theory can be used to study quantitative problems in Diophantine geometry, by focusing on the Manin conjectures, the dimension growth conjecture, and the Hardy-Littlewood circle method. . book is clear, concise, and well written, and as such is highly recommended to a beginning graduate student looking for direction in pure mathematics or number theory. . includes a number of interesting and accessible exercises at the end of each of the eight chapters."­­­ (Robert Juricevic, Mathematical Reviews, Issue 2010 i)