Title | Arithmetic of Higher-Dimensional Algebraic Varieties [electronic resource] / edited by Bjorn Poonen, Yuri Tschinkel |
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Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2004 |

Edition | 1 |

Connect to | http://dx.doi.org/10.1007/978-0-8176-8170-8 |

Descript | XVI, 287 p. online resource |

SUMMARY

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 รฉtale 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-Thรฉlรจne, 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

CONTENT

Diophantine equations: progress and problems -- Rational points and analytic number theory -- Weak approximation on algebraic varieties -- Counting points on varieties using universal torsors -- The Cox ring of a Del Pezzo surface -- Counting rational points on threefolds -- Remarques sur lโ{128}{153}approximation faible sur un corps de fonctions dโ{128}{153}une variable -- K3 surfaces over number fields with geometric Picard number one -- Jumps in Mordell-Weil rank and Arithmetic Surjectivity -- Universal torsors and Cox rings -- Random diophantine equations -- Descent on simply connected surfaces over algebraic number fields -- Rational points on compactifications of semi-simple groups of rank 1 -- Weak Approximation on Del Pezzo surfaces of degree 4 -- Transcendental Brauer-Manin obstruction on a pencil of elliptic curves

Mathematics
Algebraic geometry
Algebra
Field theory (Physics)
Functions of complex variables
Number theory
Mathematics
Number Theory
Algebraic Geometry
Field Theory and Polynomials
Several Complex Variables and Analytic Spaces