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Abelian varieties isogenous to a power of an elliptic curve. (English) Zbl 1400.14116

Given an abelian category \(\mathcal C\), an object \(E \in \mathcal C\), a ring \(R\) and a ring homomorphhism \(R \rightarrow \mathrm{End} E\), there is a natural functor \({\mathcal Hom}(-,E)\) from the opposite category of the category of finitely presented left \(R\)-modules into \(\mathcal C\). The present paper studies the functor in the case of the category \(\mathcal C\) of commutative proper group schemes over a field \(k\), and elliptic curve \(E\) over \(k\) and the ring \(R = \mathrm{End} E\). Here the images of the functor are abelian varieties isogenous to a power of \(E\). In this case there is also a functor in the reverse direction. The main result of the paper is a complete answer to the question of when the two functors are equivalences of categories. An important notion is also that of a kernel subgroup of an abelian variety \(A\), which is defined as the subgroup scheme \(\bigcap_{\alpha \in I} \ker \alpha\) of \(A\) for a left ideal \(I \subset \mathrm{End} A\). In many cases the kernel subroups of an \(A\) isogenous to a power of \(E\) are determined. Examples are given showing that the functors are not always equivalences and that not every subgroup scheme of \(A\) is a kernel subgroup. Finally there is some partial extension to higher dimensional abelian varieties.

MSC:

14K15 Arithmetic ground fields for abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
14K02 Isogeny
14K05 Algebraic theory of abelian varieties
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[1] Baker, M. H., González-Jiménez, E., González, J. and Poonen, B., Finiteness results for modular curves of genus at least 2, Amer. J. Math.127 (2005), 1325-1387. doi:10.1353/ajm.2005.0037 · Zbl 1127.11041
[2] Bass, H., On the ubiquity of Gorenstein rings, Math. Z., 82, 8-28, (1963) · Zbl 0112.26604
[3] Borevič, Z. I. and Faddeev, D. K., Integral representations of quadratic rings, Vestik Leningrad Univ.15 (1960), 52-64.
[4] Borevič, Z. I. and Faddeev, D. K., Representations of orders with a cyclic index, Proc. Steklov Inst. Math.80 (1965), 51-65; translated in Algebraic number theory and representations, ed. D. K. Faddeev (AMS, Washington, DC, 1968), pp. 56-72.
[5] Bourbaki, N., Éléments de mathématique. Algèbre. Chapitres 1 á 3, (1970), Hermann: Hermann, Paris · Zbl 0211.02401
[6] Centeleghe, T. G. and Stix, J., Categories of abelian varieties over finite fields, I: Abelian varieties over F_{p}, Algebra Number Theory9 (2015), 225-265. doi:10.2140/ant.2015.9.225 · Zbl 1395.11102
[7] Deligne, P., Variétés abéliennes ordinaires sur un corp fini, Invent. Math., 8, 238-243, (1969) · Zbl 0179.26201
[8] Eichler, M., Über die Idealklassenzahl hyperkomplexer Systeme, Math. Z., 43, 481-494, (1938) · Zbl 0018.20201
[9] Giraud, J., Remarque sur une formule de Shimura-Taniyama, Invent. Math., 5, 231-236, (1968) · Zbl 0165.54801
[10] Grothendieck, A., Techniques de construction et théorèmes d’existence en géomtrie algébrique III: préschemas quotients, Séminaire Bourbaki 13e année, 1960/61, no. 212.
[11] Kani, E., Products of CM elliptic curves, Collect. Math., 62, 297-339, (2011) · Zbl 1237.11027
[12] Lam, T. Y., Lectures on modules and rings, (1999), Springer: Springer, New York · Zbl 0911.16001
[13] Lang, S., Algebra, , revised 3rd edn (Springer, New York, 2002). doi:10.1007/978-1-4613-0041-0
[14] Lange, H., Produkte elliptischer Kurven, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 95-108, (1975) · Zbl 0317.14022
[15] Lauter, K., The maximum or minimum number of rational points on genus three curves over finite fields, with an appendix by J.-P. Serre, Compos. Math., 134, 87-111, (2002) · Zbl 1031.11038
[16] Levy, L., Modules over Dedekind-like rings, J. Algebra, 93, 1-116, (1985) · Zbl 0564.13010
[17] Li, K.-Z. and Oort, F., Moduli of supersingular abelian varieties (Springer, Berlin, 1998). doi:10.1007/BFb0095931 · Zbl 0920.14021
[18] Mestre, J.-F., La méthode des graphes. Exemples et applications, in Proceedings of the international conference on class numbers and fundamental units of algebraic number fields, Katata, 1986 (Nagoya University, Nagoya, 1986), 217-242. · Zbl 0621.14021
[19] Mumford, D., Abelian varieties, (1970), Tata Institute of Fundamental Research and Oxford University Press: Tata Institute of Fundamental Research and Oxford University Press, Oxford · Zbl 0198.25801
[20] Ogus, A., Supersingular K3 crystals, in Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II, (Société Mathématique de France, Paris, 1979), 3-86. · Zbl 0435.14003
[21] Oort, F., Which abelian surfaces are products of elliptic curves?, Math. Ann., 214, 35-47, (1975) · Zbl 0283.14007
[22] Reiner, I., Maximal orders, (2003), Oxford University Press: Oxford University Press, Oxford · Zbl 1024.16008
[23] Salce, L., Warfield domains: module theory from linear algebra to commutative algebra through abelian groups, Milan J. Math., 70, 163-185, (2002) · Zbl 1054.13005
[24] Schoen, C., Produkte abelscher Varietäten und Moduln über Ordnungen, J. Reine Angew. Math., 429, 115-123, (1992) · Zbl 0744.14030
[25] Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., 15, 259-331, (1972) · Zbl 0235.14012
[26] Serre, J.-P., Rational points on curves over finite fields, Part I: ‘\(q\) large’. Lectures given at Harvard University, September to December 1985, notes taken by Fernando Gouvêa.
[27] Shioda, T., Supersingular K3 surfaces, in Algebraic geometry, Copenhagen 1978, , ed. Lønsted, K. (Springer, Berlin, 1979), 564-591.
[28] Shioda, T. and Mitani, N., Singular abelian surfaces and binary quadratic forms, in Classification of algebraic varieties and compact complex manifolds, (Springer, Berlin, 1974), 259-287. doi:10.1007/BFb0066163
[29] Tate, J., Endomorphisms of abelian varieties over finite fields, Invent. Math., 2, 134-144, (1966) · Zbl 0147.20303
[30] Waterhouse, W. C., Abelian varieties over finite fields, Ann. Sci. Éc. Norm. Supér. (4), 2, 521-560, (1969) · Zbl 0188.53001
[31] Yu, C.-F., Superspecial abelian varieties over finite prime fields, J. Pure Appl. Algebra, 216, 1418-1427, (2012) · Zbl 1255.11031
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