By Sir Peter Swinnerton-Dyer (auth.), Björn Poonen, Yuri Tschinkel (eds.)
One of the good successes of 20th century arithmetic has been the striking qualitative realizing of rational and crucial issues on curves, gleaned partly in the course of the theorems of Mordell, Weil, Siegel, and Faltings. It has turn into transparent that the examine of rational and crucial issues has deep connections to different branches of arithmetic: advanced algebraic geometry, Galois and étale cohomology, transcendence idea and diophantine approximation, harmonic research, automorphic types, and analytic quantity theory.
This textual content, which makes a speciality of higher-dimensional types, offers accurately such an interdisciplinary view of the topic. it's a digest of analysis and survey papers through top experts; the ebook records present wisdom in higher-dimensional mathematics and provides symptoms for destiny study. will probably be invaluable 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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Additional resources for Arithmetic of Higher-Dimensional Algebraic Varieties
5J S . J . BLOCH - Higher regulators, algebraic K -theory, and zeta functions of elliptic curves, CRM Monograph Series, vol. 11, Am erican Math ematical Society, Providence, RI, 2000.  A . BOREL- Cohomologie de SL n et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977) , no. 4, 613- 636.  M. D Thesis, Cambridge Universit y, 2002. [8J M . BR IGHT & S . P . SWINNERTON-Dv ER - Computing t he Brau er-Manin obst ruct ions, to appear, 2002.  F . E.
More examples . 1. Let Q C P k be a smooth proj ective qu adric. Then Q sa tisfies weak approximation. Here we do not assume that t here is a k-rational point.
Note that this point is in th e crit ical st rip, so that the conject ure pre-supposes the ana lyt ic continuation of L 1(8, J) . At present t here are two well-understood cases in which ana lytic conti nuation is known : when K = Q , so that J can be par am etrised by means of modular functions, and when J admits complex 18 SIR PETER SW INNERTON-DYER mult ip lication. As consequence , t hese two cases ar e likely to be easier t han the gener al case; bu t even here I do not expect much furt her progress in t he next decad e.