By Sir Peter Swinnerton-Dyer (auth.), Björn Poonen, Yuri Tschinkel (eds.)

ISBN-10: 0817681701

ISBN-13: 9780817681708

ISBN-10: 1461264715

ISBN-13: 9781461264712

One of the nice successes of 20th century arithmetic has been the extraordinary qualitative figuring out of rational and critical issues on curves, gleaned partially throughout the theorems of Mordell, Weil, Siegel, and Faltings. It has turn into transparent that the learn of rational and vital issues has deep connections to different branches of arithmetic: advanced algebraic geometry, Galois and étale cohomology, transcendence conception and diophantine approximation, harmonic research, automorphic varieties, and analytic quantity theory.

This textual content, which specializes in higher-dimensional kinds, presents accurately such an interdisciplinary view of the topic. it's a digest of study and survey papers by way of major experts; the ebook records present wisdom in higher-dimensional mathematics and offers symptoms for destiny learn. it will likely be worthwhile not just to practitioners within the box, yet to a large viewers of mathematicians and graduate scholars with an curiosity in mathematics 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.

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Acknowledgements: This paper is an expanded version of two talks given at the workshop Rational and integral points on higher-dimensional varieties Key words and phrases . Weak a pproxim ati on , Br au er-Marrin obstruction , torsors, 44 DAVID HARARI (Palo-Alto , December 2002). Th e author t hanks t he AIM and the organizers of t his conference for t heir warm hospitali ty. 2. 1. Basic facts. Th e following result is a refinement of t he Chinese remainder theorem. 1 (Weak Approximation). Let E C Dk be a finite set of places of k , Let a v E k v for vEE .

A one-sided estimat e for one such famil y is given in [41]. For Del Pezzo sur faces, the valu e of c for which N(H, U) '" AH(log H) C is defined by the geomet ry rather t ha n by the numb er theory, though that is not true of A. For other varieties, the corresponding stat ement need no longer be true. We st art with curves. For a curve of genus and degree d, we have N(H, V) '" AH 2 / d ; and for a cur ve of genus great er th an 1 Faltings ' th eorem is equivalent to the st atement that N(H, V) = 0(1) .

SALBERGER & A . N . SKOROBOGATOV - Weak approximation for surfaces defined by two quadratic forms, Duke Math . J. 63 (1991), no . 2, 517-536. - P . SERRE - Facteurs locaux des fonctions zeta des varietes alqebriques (definitions et conjectures), Seminaire Delange-Pisot-Poitou, 1969/70, exp o 19. [37] C . SIEGEL - Normen algebraischer Zahlen, Werke , Band IV, Vandenhoeck & Ruprecht, 1983, 250-268 . (38) A . SILVERBERG - Open questions in arithmetic algebraic geometry, Arithmetic algebraic geometry (Park City, UT, 1999), lAS /Park City Math .

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