Abstract
An abelian variety with sufficiently many complex multiplications has potentially good reduction; in case the residue class field is finite this was proved by Serre and Tate; in this paper we give a proof in the general case. An abelian variety has potentially stable reduction (at any discrete valuation of its field of definition); we show this theorem follows directly from the Igusa-Grothendieck orthogonality theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Artin, M., Winters, G.: Degenerate fibres and stable reduction. Topology9 (1971), 373–383.
Cassels, J.W.S., Fröhlich, A.: Algebraic number theory. (Brighton conference). Thompson, Washington, 1967.
Chevalley, C.: Une démonstration d'un théorème sur les groupes algébriques. Journ. Math. pures appl.39 (1960), 307–317.
Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. No. 36 (Volume dedicated to O. Zariski), IHES, 1969, 75–109.
Fossum, R., Iversen, B.: On Picard groups of algebraic fibre spaces. To be published in Journ. Pure Appl. Algebra.
Grothendieck, A.: Le groupe de Brauer, I. Sém. Bourbaki17 (1964/65), 290 (also in: Dix exposés sur la cohomologie des schémes, North Holland Publ. Cie., Amsterdam, 1968).
Igusa, J.I.: Abstract vanishing cycle theory. Proc. Japan Acad.34 (1958), 589–593.
Kodaira, K.: On compact analytic surfaces. In: Analytic functions, Princeton Math. Series 24, Princeton Univ. Press 1960.
Mumford, D.: Geometric invariant theory. Ergebnisse der Math. Neue F. Bd. 34, Springer Verlag, 1965.
Mumford, D.: Abelian varieties. Tata Inst. Fund. Research, Studies in Math. Vol. 5, Oxford Univ. Press, 1970.
Néron, A.: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Publ. Math. No. 21, IHES 1964.
Raynaud, M.: Modèles de Néron. C.R. Acad. Sc. Paris262, Sér. A, 345–347 (7 fév. 1966).
Raynaud, M.: Faisceaux amples sur les schémas en groupes et les espaces homogènes. Lect. Notes Math. 119, Springer Verlag, 1970.
Raynaud, M.: Compactification du module des courbes. Sém. Bourbaki23 (1970/1971), No. 385.
Séminaire de Géom. algébrique, SGA 7 I: Groupes de monodromie en géométrie algébrique. Lecture Notes in Math. 288, Springer Verlag, 1972.
Serre, J.-P.: Corps locaux. Act. Sc. Ind. 1296. Hermann, Paris, 1962.
Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. of Math.88 (1968), 492–517.
Shimura, G., Taniyama, Y.: Complex multiplication of abelian varieties. Publ. Math. Soc. Japan, Vol. 6, 1961.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Oort, F. Good and stable reduction of abelian varieties. Manuscripta Math 11, 171–197 (1974). https://doi.org/10.1007/BF01184956
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01184956