Arithmetic of Higher-Dimensional Algebraic Varieties

✍ Scribed by Sir Peter Swinnerton-Dyer (auth.), Björn Poonen, Yuri Tschinkel (eds.)

• Publisher: Birkhäuser Basel
• Year: 2004
• Tongue: English
• Leaves: 292
• Series: Progress in Mathematics 226
• Edition: 1
• Category: Library

✦ Synopsis

One of the great successes of twentieth century mathematics has been the remarkable qualitative understanding of rational and integral points on curves, gleaned in part through the theorems of Mordell, Weil, Siegel, and Faltings. It has become clear that the study of rational and integral points has deep connections to other branches of mathematics: complex algebraic geometry, Galois and étale cohomology, transcendence theory and diophantine approximation, harmonic analysis, automorphic forms, and analytic number theory.

This text, which focuses on higher-dimensional varieties, provides precisely such an interdisciplinary view of the subject. It is a digest of research and survey papers by leading specialists; the book documents current knowledge in higher-dimensional arithmetic and gives indications for future research. It will be valuable not only to practitioners in the field, but to a wide audience of mathematicians and graduate students with an interest in arithmetic geometry.

Contributors: Batyrev, V.V.; Broberg, N.; Colliot-Thélène, J-L.; Ellenberg, J.S.; Gille, P.; Graber, T.; Harari, D.; Harris, J.; Hassett, B.; Heath-Brown, R.; Mazur, B.; Peyre, E.; Poonen, B.; Popov, O.N.; Raskind, W.; Salberger, P.; Scharaschkin, V.; Shalika, J.; Starr, J.; Swinnerton-Dyer, P.; Takloo-Bighash, R.; Tschinkel, Y.: Voloch, J.F.; Wittenberg, O.

✦ Table of Contents

Front Matter....Pages i-xvi
Front Matter....Pages 1-1
Diophantine Equations: Progress And Problems....Pages 3-35
Rational Points and Analytic Number Theory....Pages 37-42
Weak Approximation on Algebraic Varieties....Pages 43-60
Counting Points On Varieties Using Universal Torsors....Pages 61-81
Front Matter....Pages 83-83
The Cox Ring of a Del Pezzo Surface....Pages 85-103
Counting Rational Points On Threefolds....Pages 105-120
Remarques Sur L’Approximation Faible Sur Un Corps De Fonctions D’Une Variable....Pages 121-134
K3 Surfaces Over Number Fields with Geometric Picard Number One....Pages 135-140
Jumps in Mordell-Weil Rank and Arithmetic Surjectivity....Pages 141-147
Universal Torsors and Cox Rings....Pages 149-173
Random Diophantine Equations....Pages 175-184
Descent on Simply Connected Surfaces Over Algebraic Number Fields....Pages 185-204
Rational Points on Compactifications of Semi-Simple Groups of Rank 1....Pages 205-233
Weak Approximation on Del Pezzo Surfaces of Degree 4....Pages 235-257
Transcendental Brauer-Manin Obstruction on a Pencil Of Elliptic Curves....Pages 259-267
Back Matter....Pages 269-287

✦ Subjects

Number Theory; Algebraic Geometry; Field Theory and Polynomials; Several Complex Variables and Analytic Spaces