Synopses & Reviews
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.
These articles which are written by leading experts make interesting reading and also give the non expert reader an idea of the subject.
This text offers a collection of survey and research papers by leading specialists in the field documenting the current understanding of higher dimensional varieties. Recently, it has become clear that ideas from many branches of mathematics can be successfully employed in the study of rational and integral points. This book will be very valuable for researchers from these various fields who have an interest in arithmetic applications, specialists in arithmetic geometry itself, and graduate students wishing to pursue research in this area.
Table of Contents
Abstracts.- Introduction.- Part I. Expository Articles. Swinnerton-Dyer, P.: Diophantine equations: progress and problems. Heath-Brown, R.: Rational points and analytic number theory. Harari, D.: Weak approximation on algebraic varieties. Peyre, E.: Counting points on varieties using universal torsors.- Part II. Research Articles. Batyrev, V.V.; Popov, O.N.: The Cox ring of a Del Pezzo surface. Broberg, N.; Salberger, P.: Counting rational points on threefolds. Colliot-Thélène, J-L.; Gille, P.: Remarques sur l'approximation faible sur un corps de fonctions d'une variable. Ellenberg, J.S.: K3 surfaces over number fields with geometric Picard number one. Graber, T.; Harris, J.; Mazur, B.; Starr, J.: Jumps in Mordell-Weil rank and Arithmetic Surjectivity. Hassett, B.; Tschinkel, Y.: Universal torsors and Cox rings. Poonen, B.; Voloch, J.F.: Random diophantine equations. Raskind, W.; Scharaschkin, V.: Descent on simply connected surfaces over algebraic number fields. Shalika, J.; Takloo-Bighash, R.; Tschinkel, Y.: Rational points on compactification of semi-simple groups of rank 1. Swinnerton-Dyer, P.: Weak Approximation on Del Pezzo surfaces of degree 4. Whittenberg, O.: Transcendental Brauer-Manin obstruction on a pencil of elliptic curves. - Glossary - Index