Download Arithmetic Algebraic Geometry: Lectures given at the 2nd by Jean-Louis Colliot-Thelene, Kazuya Kato, Paul Vojta, Edoardo PDF

Algebraic Geometry

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

ISBN-10: 3540571108

ISBN-13: 9783540571100

This quantity includes 3 lengthy lecture sequence by way of J.L. Colliot-Thelene, Kazuya Kato and P. Vojta. Their themes are respectively the relationship among algebraic K-theory and the torsion algebraic cycles on an algebraic kind, a brand new method of Iwasawa thought for Hasse-Weil L-function, and the purposes of arithemetic geometry to Diophantine approximation. They include many new effects at a truly complex point, but in addition surveys of the cutting-edge at the topic with whole, special profs and many historical past. accordingly they are often priceless to readers with very various history and adventure. CONTENTS: J.L. Colliot-Thelene: Cycles algebriques de torsion et K-theorie algebrique.- ok. Kato: Lectures at the method of Iwasawa concept for Hasse-Weil L-functions.- P. Vojta: functions of mathematics algebraic geometry to diophantine approximations.

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Additional 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

On analyse le groupe Haa(X,Q/Z(2)) au moyen de la suite spectrale de Hochschild-Serre. Le groupe H3(k, Q/Z(2)) est nul (k est local, donc de dimension cohomologique ca(k) = 2). 2 (d) ci-dessus). 3, l'hypoth~se H'(X, Ox) = 0 garantit la nullitg du sous-groupe divisible maximal de g ~ t ( ~ , Q/Z(2)), ce qui assure £ peu de frais que le groupe g2(k, H~t(X , Q/Z(2))) est d'exposant fini). Or le groupe H~(k, Hlt(-X,Q/Z(2))) est ici fini, car l'hypoth&se de bonne r~duction de la vari~t~ d'Albanese, donc de la vari~t~ de Picard, sur une extension firSe K de k assure que le groupe H2(k,H~t('K,Q/I(2)) °) est nul (cf.

Dont nous avons 6tabli qu'il est un quotient du pr6c6dent. Mais, d'apr6s le d6but de la d6monstration, le groupe CH2(X)I_to,, est d'exposant fini. C'est donc un groupe fini. [7 47 BIBLIOGRAPHIE [BV] L. - - Cohomology theories and algebraic cycles on singular varieties, Contemp. Math. 126 (1992) 197-217; Theorie coomologiche e cicli aigebricl sulle varieta' singolari, Tesi di dottorato, Universita' di Genova (1991). [B1] S. - - Torsion algebraic cycles and a theorem of Roitman, Comp. Math. 39 (1979) 107-127.

En l'absence de point rationnel, il est cependant clair que l'image de N dans k* est d'indice fini, car c'est d~j£ le cas pour chaque Np, k 6tant un corps local. On peut se demander si N n'est pas toujours surjective, m6me en l'absence de point k-rationnel. Pour certaines surfaces, l'approche de Saito permet de contr61er le groupe CH2(X)tor, -CHo(X)tor, au moyen d'applications cycles, et de faqon peut-Stre plus frappante encore, au moyen du groupe de Brauer. Le th~orbme suivant s'applique en paxticulier aux surfaces avec H2(X, O x ) = 0 et H i ( x , O x ) = 0 (cette derai~re condition assurant la triviaiit~ de la vaxi~t~ d'Albanese), en particulier aux surfaces d'Enriques et aux surfaces rationnelles.

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