×

zbMATH — the first resource for mathematics

The isomorphism classes of abelian varieties of CM-type. (English) Zbl 1087.14030
It is well known (Grothendieck, Oort) that an abelian variety of CM-type over a field \(k\) of positive characteristic is isogenous (over a finite extension) to an abelian variety defined over a finite field. A more precise criterion is obtained in this paper. The author shows that in positive characteristic, a CM abelian variety with CM field \(L\) is defined over a finite field if its endomorphism ring contains \(O_L\), the ring of integers of \(L\). Based on this, a classification of the isomorphism classes of CM abelian varieties is given, and a criterion is obtained for a polarized abelian variety with CM by \(O_L\) to be liftable to characteristic \(0\).

MSC:
14K22 Complex multiplication and abelian varieties
11G15 Complex multiplication and moduli of abelian varieties
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Deligne, P.; Pappas, G., Singularités des espaces de modules de Hilbert, en LES caractéristiques divisant le discriminant, Compositio math., 90, 59-79, (1994) · Zbl 0826.14027
[2] Kottwitz, R.E., Isocrystals with additional structure, Composito. math., 56, 201-220, (1985) · Zbl 0597.20038
[3] Kottwitz, R.E., Points on some Shimura varieties over finite fields, J. amer. math. soc., 5, 373-444, (1992) · Zbl 0796.14014
[4] Lang, S., Abelian varieties, (1959), Interscience New York
[5] Lang, S.; Néron, A., Rational points of abelian varieties over function fields, Amer. J. math., 81, 95-118, (1959) · Zbl 0099.16103
[6] Li, K.-Z.; Oort, F., Moduli of supersingular abelian varieties, Lecture notes in mathematics, Vol. 1680, (1998), Springer Berlin · Zbl 0920.14021
[7] Manin, Yu., Theory of commutative formal groups over fields of finite characteristic, Russian math. surveys, 18, 1-80, (1963) · Zbl 0128.15603
[8] Mumford, D., Abelian varieties, (1974), Oxford University Press Oxford · Zbl 0199.24601
[9] Mumford, D.; Fogarty, J.; Kirwan, F., Geometric invariant theory, (1994), Springer Berlin · Zbl 0797.14004
[10] Oort, F., The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field, J. pure appl. algebra, 3, 399-408, (1973) · Zbl 0272.14009
[11] Pappas, G., On the arithmetic moduli schemes of PEL Shimura varieties, J. algebraic geom., 9, 577-605, (2000) · Zbl 0978.14023
[12] M. Rapoport, Th. Zink, Period Spaces for p-divisible groups, Annals of Mathematics Studies, 141, Princeton University Press, 1996. · Zbl 0873.14039
[13] Reimann, H.; Zink, Th., Der dieudonnémodul einer polarisierten abelschen mannigfaltigkeit vom CM-typ, Ann. math., 128, 461-482, (1988) · Zbl 0674.14030
[14] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Society, Japan, Vol. 11, Iwanami Shoten, Tokyo, Princeton University Press, Princeton, NJ, 1971. · Zbl 0221.10029
[15] Serre, J.-P.; Tate, J., Good reduction of abelian varieties, Ann. math., 88, 492-517, (1968) · Zbl 0172.46101
[16] Tate, J., Endomorphisms of abelian varieties over finite fields, Invent. math., 2, 134-144, (1996) · Zbl 0147.20303
[17] Tate, J., Classes d’isogenie de variétés abéliennes sur un corps fini (d’après T. honda). Sém. bourbaki exp. 352 (1968/69), Lecture notes in mathematics, Vol. 179, (1971), Springer Berlin
[18] C.-F. Yu, On the supersingular locus in Hilbert-Blumenthal 4-folds, J. Algebraic Geom. 12 (2003) 653-698. · Zbl 1059.14031
[19] Yu, C.-F., Lifting abelian varieties with additional structures, Math. Z., 242, 427-441, (2002) · Zbl 1051.14052
[20] T. Zink, Cartiertheorie kommutativer formaler Gruppen. Teubner-Texte Math. Teubner, Leipzip, 1984. · Zbl 0578.14039
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.