×

Rational points of Abelian varieties with values in towers of number fields. (English) Zbl 0245.14015


MSC:

14G25 Global ground fields in algebraic geometry
11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
11R23 Iwasawa theory

References:

[1] Artin, M.: Auto-duality of the Jacobian. Mineographed notes. Bowdoin Summer Conference in Algebraic Geometry, 1967.
[2] [GT] Artin, M.: Grothendieck topologies. Mimeographed notes. Harvard University, 1962.
[3] [SGAA] Artin, M., Grothendieck, A.: Séminaire de géometrie algébrique 1963-64. Cohomologie, étale des schémas Mimeographed notes. Institut des Hautes Etudes Scientifiques, Paris.
[4] Artin, M., Mazur, B.: Etale homotopy. Lecture Notes in Math. no. 178. Berlin-Heidelberg-New York: Springer 1969. · Zbl 0182.26001
[5] Artin, M., Mazur, B.: Flat arithmetic duality (in preparation). · Zbl 0188.53402
[6] Artin, M., Verdier, J.L.: Etale arithmetic duality. Proceedings of the summer conference in Algebraic Geometry held at Woodshole, Mass. 1965.
[7] Birch, B.J., Swinnerton-Dyer, H.P.F.: Notes on elliptic curves I. J. Reine Angew. Math.212, 7-25 (1963); 11218, 79-108 (1965). · Zbl 0118.27601 · doi:10.1515/crll.1963.212.7
[8] Burnside, W.: The theory of groups (2nd Ed.). Cambridge University Press 1911. · JFM 42.0151.02
[9] Cassels, J. W. S.: On a diophantine equation. Acta Arithmetica6, 47-52 (1960). · Zbl 0094.25701
[10] Cassels, J. W. S.: Diophantine equations with special reference to elliptic curves. J. London Math. Soc.41, 193-291 (1966). · Zbl 0138.27002 · doi:10.1112/jlms/s1-41.1.193
[11] Cassels, J. W. S.: Arithmetic on curves of genus one (IV). J. Reine Angew. Math.211, 95-112 (1962). · Zbl 0106.03706 · doi:10.1515/crll.1962.211.95
[12] Cassels, J. W. S., Fröhlich, A. (eds.): Algebraic number theory. London-New York: Academic Press 1967. · Zbl 0153.07403
[13] Cassels, J. W. S., Sansone G.: Sur le probleme de M. Werner Mnich. Acta Arithmetica7, 187-190 (1961/62).
[14] Deligne, P.: Variétés abéliennes ordinaires sur un corps fini. Inventiones math.8, 238-243 (1969). · Zbl 0179.26201 · doi:10.1007/BF01406076
[15] [SGAD] Demazure, M., Grothendieck, A.: Schémas en groupes. Séminaire I. H. E. S., 1963-64. Lecture Notes in Math. nos. 151-153. Berlin-Heidelberg-New York: Springer 1970.
[16] Eichler, M.: Quaternäre, quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion. Arch. Math.5, 355-366 (1954). · Zbl 0059.03804 · doi:10.1007/BF01898377
[17] Greenberg, M. J.: Schemata over local rings. Ann. of Math.73, no. 3, 624-648 (1961); II, Ann. of Math.78, no. 2, 256-266 (1963). · Zbl 0115.39004 · doi:10.2307/1970321
[18] Greenberg, M. J.: Pro-algebraic structure on the rational subgroup, of ap-adic abelian variety. Ph. D. thesis. Princeton University 1959.
[19] Grothendieck, A.: Sur quelques points d’algèbre homologique. Tohoku Math. J.9, 119-221 (1957). · Zbl 0118.26104
[20] [GB III] Grothendieck, A.: Le groupe de Brauer III: exemples et compléments (a continuation of Bourbaki exposés: 290, 297). Published in Dix exposés sur la cohomologie des schémas. Amsterdam: North-Holland Pub. Cie. 1968.
[21] Grothendieck, A.: Techniques de déscente et théorèmes d’existence en géometrie algébrique. Séminaire Bourbaki, 12, exp. 195 (1959-60). New York-Amsterdam: Benjamin Inc. 1966.
[22] [EGA] Grothendieck, A.: Redigé avec la collaboration de J. Dieudonné, Éléments de géométrie algébrique. Publications Mathematiques, I. H. E. S., 4, 8, 11, 17, 20, 24, 28, 32 Paris (1961-68).
[23] [SGA] Grothendieck A.: Revêtements étales et groupes fondamentaux. Séminaires de Géométrie Algébrique à l’I. H. E. S. (60-61). Lecture Notes in Math. no. 224. Berlin-Heidelberg-New York: Springer 1971.
[24] Hardy, G. H., Littlewood, J. E.: Some problems of partitic numerorum III. Acta Math.44, 1-70 (1923); reprinted in: G. H. Hardy, Collected papers, vol. 1, Oxford (1966), 561-630. · JFM 48.0143.04 · doi:10.1007/BF02403921
[25] Hasse, H.: Existenz separabler zyklischer unverzweigter Erweiterungskörper vom Primzahlgradep über elliptischen Funktionenkörpern der Charkteristikp. J. Reine Angew. Math.172, 2, 77-85 (1934). · JFM 60.0910.02
[26] Hasse, H., Witt, E.: Zyklischer unverzweigter Erweiterungskörper vom Primzahlgradep über einem Funktionenkörper der Charakteristikp. Monatshefte für Math. u. Physik43, 477-492 (1936). · Zbl 0013.34102 · doi:10.1007/BF01707628
[27] Honda, T.: Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan20, 83-95 (1968). · Zbl 0203.53302 · doi:10.2969/jmsj/02010083
[28] Igusa, J.: Kroneckerian model of fields of elliptic modular functions. Amer. J. Math.81, 561-577 (1959). · Zbl 0093.04502 · doi:10.2307/2372914
[29] Iwasawa, K.: On some properties of ?-finite modules. Ann. of Math.70, no. 2, 291-312 (1959). · Zbl 0202.33102 · doi:10.2307/1970105
[30] Iwasawa, K.: On ?-extensions of number fields. Bull. Amer. Math. Soc.65, no. 4, 183-226 (1959). · Zbl 0089.02402 · doi:10.1090/S0002-9904-1959-10317-7
[31] Iwasawa, K.: On the theory of cyclotomic fields. Ann. of Math.70, no. 3, 530-561 (1959). · Zbl 0093.04403 · doi:10.2307/1970328
[32] Iwasawa, K., Sims, C. C.: Computation of invariants in the theory of cyclotomic fields. J. of the Math. Soc. of Japan18, no. 1, 86-96 (1966). · Zbl 0141.04901 · doi:10.2969/jmsj/01810086
[33] Kubota, T., Leopoldt, H. W.: Einep-adische Theorie der Zetawerte (Teil I) J. Reine Angew. Math.213, 228-239 (1964).
[34] Lang, S.: Algebraic numbers. Reading, Mass: Addison-Wesley 1964. · Zbl 0211.38501
[35] Lang, S.: Algebraic groups over finite fields. Amer. J. Math.78, no. 3, 555-563 (1956). · Zbl 0073.37901 · doi:10.2307/2372673
[36] Ligozat, G.: FonctionL des courbes modulaires. Mimeo. notes. Séminaire Delange-Pisot-Poitou, 1969/70, no. 9. Version to appear Courbes modulaires de genre 1.
[37] Manin, Ju.: Cyclotomic fields and modular curves [in Russian]. Uspekhi Mat. Nauk. Tom XXVI6, (162), 7-71 (1971). Translation to appear in Russian Math. Surveys. London Math. Society.
[38] Mazur, B.: Rational points of Abelian varieties with values in towers of number fields. Mimeo. notes, Harvard U. 1969.
[39] Mazur, B.: Arithmétique des courbes elliptiques sur les corps cyclotomiques. Mimeographed notes by J. F. Boutot of a course given at Orsay 1970, distributed by I. H. E. S. Paris.
[40] Mazur, B.: Local flat duality Amer. Journal of Math.92, 343-361 (1970). · Zbl 0199.24501 · doi:10.2307/2373327
[41] Mazur, B., Roberts, L.: Local Euler characteristics. Inventiones math.9, 201-234 (1970). · Zbl 0191.19202 · doi:10.1007/BF01404325
[42] Mazur, B., Swinnerton-Dyer H. P. F.: Thep-adicL-series of an elliptic curve (in preparation).
[43] Milne, J. S.: Extensions of abelian varieties defined over a finite field. Inventiones math.5, 63-84 (1968). · Zbl 0205.24901 · doi:10.1007/BF01404538
[44] Mumford, D.: Lectures on curves on an algebraic surface (with the assistance of G. M. Bergman). Ann. of Math. Studies59, Princeton, 1966. · Zbl 0187.42701
[45] Mumford, D.: Geometric invariant theory. Ergebnisse Math., Bd. 34. Berlin-Heidelerg-New York: Springer 1965. · Zbl 0147.39304
[46] Mumford, D., Oort, F.: Deformations and lifting of finite commutative group schemes. Inventiones math.5, 317-334 (1968). · Zbl 0179.49901 · doi:10.1007/BF01389779
[47] Néron, A.: Modèles mimimaux des variétés abéliennes sur les corps locaux et globaux. Publications Mathematiques, I. H. E. S., no. 21 (1964).
[48] Ogg, A.: Elliptic curves and wild ramification. Amer. J. of. Math.89, 1-21 (1967). · Zbl 0147.39803 · doi:10.2307/2373092
[49] Oort, F.: Commutative group schemes. Lecture Notes in Mathematics no. 15. Berlin-Heidelberg-New York: Springer 1966. · Zbl 0216.05603
[50] Oort, F., Tate, J.: Group schemes of prime order. Ann. Scient. Ec. Norm. Sup., series 4,3, 1-21 (1970). · Zbl 0195.50801
[51] Raynaud, M.: Passage au quotient par une relation d’équivalence plate. Proc. of a Conference on Local Fields. Berlin-Heidelberg-New York: Springer 1967. · Zbl 0165.24003
[52] Serre, J.-P.: Classes de corps cyclotomiques. Séminie bourbaki no. 174 (1958). New York-Amsterdam: W. A. Benjamin, Inc. 1966.
[53] [CG] Serre, J.-P.: Cohomologie Galoisienne. Lecture Notes in Mathematics no. 5. Berlin-Heidelberg-New York: Springer 1964.
[54] [CL] Serre, J.-P.: Corps locaux. Paris: Hermann 1962.
[55] Serre, J.-P.: Sur les corps locaux à corps résidue algébriquement clos, 2. Bull. Soc. Math. France89, 105-154 (1961). · Zbl 0166.31103
[56] Serre, J.-P.: Corps locaux et isogénies. Séminaire Bourbaki, exposé 185, 1958-59.
[57] Serre, J.-P.: Groupes proalgébriques, I.H.E.S., Publication Mathematique no. 7 (1960).
[58] Serre, J.-P.: Groupes de Liel-adiques attachés aux courbes elliptiques. Colloque de Clermont-Ferrand (1964). Mimeographed notes published by I.H.E.S.
[59] [LG] Serre, J.-P.: Lie algebra and Lie groups. Lectures at Harvard University, 1964. New York-Amsterdam: W. A. Benjamin Inc. 1965.
[60] Serre, J.-P.: Abelianl-adic representations and elliptics curves. Lectures at McGill University. New York-Amsterdam: W. A. Benjamin Inc. 1968.
[61] Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Inventiones math15, 259-331 (1972). · Zbl 0235.14012 · doi:10.1007/BF01405086
[62] Serre, J.-P.: Groupesp-divisibles. Séminaire Bourbaki, exp. 318 (1966-67). New York-Amsterdam: W. A. Benjamin Inc. 1966.
[63] Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. of Math.88, 492-517 (1968). · Zbl 0172.46101 · doi:10.2307/1970722
[64] Shimura, G.: Correspondances modulaires et les fonctions zêta de courbes algébriques. J. Math. Soc. Japan10, 1-28 (1958). · Zbl 0081.07603 · doi:10.2969/jmsj/01010001
[65] Shimura, G., Taniyama, Y.: Complex multiplication of abelian varietie and its applications to number theory. Publ. Math. Soc., Japan, no. 6, Tokyo 1961. · Zbl 0112.03502
[66] Swinnerton-Dyer, P.: The conjectures of Birch and Swinnerton-Dyer and of Tate. Proceeding of a conference on Local Fields, NUFFIC Summer School helds at Driebergen in 1966, p. 132-157. Berlin-Heidelberg-New York: Springer 1967.
[67] Tate, J.: Duality theorems in Galois cohomology over number fields. Proc. Intern. Congress Math., at Stockholm, 1962, 288-295. Institute Mittag-Leffler Djursholm, Sweden (1963).
[68] Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki, exp. 306 (1966). New York-Amsterdam: W. A. Benjamin Inc. 1966. · Zbl 0199.55604
[69] Tate, J.:p-divisible groups. Proceedings of a Conference on Local Fields, NUFFIC Summer School held at Driebergen, p. 158-183 (1966). Berlin-Heidelberg-New York: Springer 1967.
[70] Tate, J.: Classes d’isogénie des variétés abéliennes sur un corps fini, (d’après T. Honda). Séminaire Bourbaki, exp. 352 (1968-69).
[71] Tate, J.: Endomorphisms of abelian varieties over finite fields. Inventiones math.2, 134-144 (1966). · Zbl 0147.20303 · doi:10.1007/BF01404549
[72] Weil, A.: Varietés abéliennes et courbes algébriques. Paris: Hermann 1948.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.