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Get Arithmetic Algebraic Geometry: Lectures given at the 2nd PDF

By Jean-Louis Colliot-Thelene, Kazuya Kato, Paul Vojta, Edoardo Ballico

ISBN-10: 3540479090

ISBN-13: 9783540479093

ISBN-10: 3540571108

ISBN-13: 9783540571100

This quantity comprises 3 lengthy lecture sequence through J.L. Colliot-Thelene, Kazuya Kato and P. Vojta. Their issues are respectively the relationship among algebraic K-theory and the torsion algebraic cycles on an algebraic style, a brand new method of Iwasawa idea for Hasse-Weil L-function, and the functions of arithemetic geometry to Diophantine approximation. They include many new effects at a truly complex point, but in addition surveys of the state-of-the-art at the topic with entire, specific profs and many historical past. therefore they are often worthwhile to readers with very assorted historical past and event. CONTENTS: J.L. Colliot-Thelene: Cycles algebriques de torsion et K-theorie algebrique.- ok. Kato: Lectures at the method of Iwasawa thought for Hasse-Weil L-functions.- P. Vojta: purposes of mathematics algebraic geometry to diophantine approximations.

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Extra resources for Arithmetic Algebraic Geometry: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Trento, Italy, June 24–July 2, 1991

Example text

Soit X une vari~t~ projective, li,~e, gdomdtriquement conneze ,ur un corpa de nombres k. Suppo,on, le groupe H2(X, Ox ) -- O. Alor~ le $ous-groupe de torsion CH2(X)tor, eat fini. 6, le groupe CH2(X)to~, est d'exposant fini. Pour presque tout premier l, la partie /-primMre CH2(X)t_tor, de CH2(X)to~, est donc nulle, et pour 6tablir le th6or~me, il suffit d'fitablir la finitude de CH2(X)t_to~ pour tout premier I. 3, on considbre la suite spectrale ell cohomologie ~tale HP(k, Hq**(X,Q t / l t ( 2 ) ) ) ==~ H2t(X, Q,/Zz(2)), qui donne lieu £ une filtration F ° C F 1 C F 2 C F 3 sur le groupe Ha,t(X, qt/Z~(2)).

Le th6orbme suivant avait 6t6 obtenu sous des hypoth6ses plus restrictives darts [CR1]. 3 ci-dessous qui permet d'aller un peu plus loin, en 61iminant les hypothbses du type : H i ( X , COz) = 0, ou X (ou Albx) a bonne r6duction. 2. Soient k un corps local, eztension finie de Qp, et X une k-varidt~ projective, lisse et gdom~triquement int~gre. Supposons H2( X, O x ) = O. Alors : (a) Le groupe ker[Cg2(x) ~ CH2(X)] - - ° " eat fini. (b) Sf de pI,,s l'i,~age d*, groupe NH~,(X, Q/Z(2)) dons le gro~,~,e H~,(X, Q/Z(2)) est fini, par ezemple ai Ha~,(-K,Q / l ( 2 ) ) c est fini, slots le groupe CH2(X)tor, est fini.

Supposons H2(X, O x ) = O. Soil n > 0 un entier naturel. Alor~ le sou~-groupe de n-torsion , C H 2 ( X ) e~t fini. D~monstration : a) I1 existe une extension finie de corps L / k teUe que X ( L ) ~ 0 et que l'application naturelle de groupes de type fini N S ( X L ) ~ NS(-'X) soit un isomorphisme. Soit m = [L : k]. e. B est principal), et que (via le th6or~me de semi-eontinuit6 de Grothendieek) H2(Xp,Oxp) = 0 pour ehaeune des fibres Xp au-dessus d'un point ferm6 p E Spee(A), le m~me 6none6 valant alors aussi pour les fibres du seh6ma XB/Spec(B) aux points ferm6s q E Spec(B).

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Arithmetic Algebraic Geometry: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Trento, Italy, June 24–July 2, 1991 by Jean-Louis Colliot-Thelene, Kazuya Kato, Paul Vojta, Edoardo Ballico


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