Number Theory and Algebraic Geometry

doi:10.1017/CBO9780511734946

Sir Peter Swinnerton-Dyer's mathematical career encompasses more than 60 years' work of amazing creativity. This volume provides contemporary insight into several subjects in which Sir Peter's influence has been notable, and is dedicated to his 75th birthday. The opening section reviews some of his many remarkable contributions to mathematics and other fields. The remaining contributions come from leading researchers in analytic and arithmetic number theory, and algebraic geometry. The topics treated include: rational points on algebraic varieties, the Hasse principle, Shafarevich-Tate groups of elliptic curves and motives, Zagier's conjectures, descent and zero-cycles, Diophantine approximation, and Abelian and Fano varieties.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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In Lieu of Birthday Greetings
pp 1-22
- By B. J. Birch, Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB, UK, Jean-Louis Colliot-Thélène, C.N.R.S., U.M.R. 8628, Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay, France, G. K. Sankaran, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, England, Miles Reid, Math Inst., Univ. of Warwick, Coventry CV4 7AL, England, Alexei Skorobogatov, Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, England
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Peter Swinnerton-Dyer's mathematical papers to date
pp 23-30
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On the Hasse principle for bielliptic surfaces
pp 31-40
- By Carmen Laura Basile, Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, England, Alexei Skorobogatov, Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, England
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Effective Diophantine approximation on Gm
pp 41-62
- By Enrico Bombieri, School of Mathematics, Institute for Advanced Study, Princeton, N.J. 08540, USA, Paula B. Cohen, Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
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A Diophantine system
pp 63-68
- By Andrew Bremner, Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, USA
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Valeurs d'un polynôme à une variable représentées par une norme
pp 69-90
- By J.-L. Colliot-Thélène, C.N.R.S., U.M.R. 8628, Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay, France, D. Harari, D.M.A., E.N.S., 45 rue d'Ulm, F-75005 Paris, France, A. N. Skorobogatov, Department of Mathrmsyics, Imperial College, 180 Queen' Gate, London SW7 2BZ, United Kingdom
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Constructing elements in Shafarevich–Tate groups of modular motives
pp 91-118
- By Neil Dummigan, University of Sheffield, Department of Pure Mathematics, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, U.K., William Stein, Harvard University, Department of Mathematics, One Oxford Street, Cambridge, MA 02138, U.S.A., Mark Watkins, Penn State Mathematics Department, University Park, State College, PA 16802, U.S.A.
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A counterexample to a conjecture of Selmer
pp 119-132
- By Tom Fisher, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
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Linear relations amongst sums of two squares
pp 133-176
- By D.R. Heath-Brown, Mathematical Institute, 24-29 St Giles', Oxford OX1 3LB, England
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Kronecker double series and the dilogarithm
pp 177-202
- By Andrey Levin, L. D. Landau Institute of Theoretical Physics, Russian Academy of Sciences, 117940 Moscow, Russia
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On Shafarevich–Tate groups and the arithmetic of Fermat curves
pp 203-226
- By William G. McCallum, Department of Mathematics, P.O. Box 210089, 617 N. Santa Rita, The University of Arizona, Tucson, AZ 85721-0089, USA, Pavlos Tzermias, Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA
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Cascades of projections from log del Pezzo surfaces
pp 227-250
- By Miles Reid, Math Inst., Univ. of Warwick, Coventry CV4 7AL, England, Kaori Suzuki, Graduate School of Mathematical Sciences, University of Tokyo 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
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On obstructions to the Hasse principle
pp 251-278
- By Per Salberger, Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden
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Abelian surfaces with odd bilevel structure
pp 279-300
- By G.K. Sankaran, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, England
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Number Theory and Algebraic Geometry

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

In Lieu of Birthday Greetings
pp 1-22

Peter Swinnerton-Dyer's mathematical papers to date
pp 23-30

On the Hasse principle for bielliptic surfaces
pp 31-40

Effective Diophantine approximation on Gm
pp 41-62

A Diophantine system
pp 63-68

Valeurs d'un polynôme à une variable représentées par une norme
pp 69-90

Constructing elements in Shafarevich–Tate groups of modular motives
pp 91-118

A counterexample to a conjecture of Selmer
pp 119-132

Linear relations amongst sums of two squares
pp 133-176

Kronecker double series and the dilogarithm
pp 177-202

On Shafarevich–Tate groups and the arithmetic of Fermat curves
pp 203-226

Cascades of projections from log del Pezzo surfaces
pp 227-250

On obstructions to the Hasse principle
pp 251-278

Abelian surfaces with odd bilevel structure
pp 279-300

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Number Theory and Algebraic Geometry

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