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Principally polarized ordinary abelian varieties over finite fields. (English) Zbl 0859.14016
Let \(k\) be a finite field of characteristic \(p\). An abelian variety \(A\) over \(k\) is ordinary if the \(p\)-rank of its group of geometric \(p\)-torsion points in equal to its dimension. P. Deligne [“Variétés abéliennes ordinaires sur un corps funi”, Invent. Math. 8, 238-243 (1969; Zbl 0179.26201)] showed that the category of ordinary abelian varieties over \(k\) is equivalent to a category whose objects are pairs \((T,F)\), where \(T\) is a free finite-rank \(\mathbb{Z}\)-module and \(F\) is an endomorphism of \(T\) that satisfies certain conditions; if one chooses an embedding \(\varphi:W\to \mathbb{C}\) of the Witt vectors over \(k\) into the complex numbers, then the \(T\) associated to a given \(A\) is the first integral homology group of the complex abelian variety obtained via \(\varphi\) from the Serre-Tate canonical lift of \(A\), and the \(F\) associated to \(A\) is the endomorphism of \(T\) obtained from the lift of the Frobenius of \(A/k\). One can view Deligne’s category equivalence as a dictionary that translates concepts from the category of ordinary abelian varieties over \(k\) to the category of these “Deligne modules” \((T,F)\).
In the first part of the present paper, we add a few entries to Deligne’s dictionary by defining polarizations of Deligne modules and by showing how the group scheme structure of the kernel of an isogeny of abelian varieties can be determined from the corresponding isogeny of Deligne modules.
In the second part of the paper we address the following questions: Given an isogeny class \({\mathcal C}\) of ordinary abelian varieties over \(k\), how can one tell which group schemes occur as the kernels of polarizations of varieties in \({\mathcal C}\)? And more specifically, how can one tell whether \({\mathcal C}\) contains a principally polarized variety? These questions become more tractable when rephrased in terms of Deligne modules. We find that there is an element of an obstruction group that determines the finite group schemes that occur as kernels of polarizations of varieties in \({\mathcal C}\), up to semi-simplification. We show how the group and the element can be calculated from the characteristic polynomial of Frobenius of the varieties in \({\mathcal C}\), and using this result we show that every simple odd-dimensional ordinary abelian variety over a finite field is isogenous to a principally polarized variety. [This last result remains true even without the “ordinary” hypothesis; see the author, “Kernels of polarizations of abelian varieties over finite fields”, J. Algebr. Geom. 5, No. 3, 583-608 (1996).]
To demonstrate the use of our techniques, in the third part of the paper we determine the characteristic polynomials of Frobenius of the isogeny classes of two-dimensional ordinary abelian varieties over a finite field that do not contain principally polarized varieties [compare to H.-G. Rück, Compos. Math. 76, No. 3, 351-366 (1990; Zbl 0742.14037)], and we produce the Weil numbers of several simple four-dimensional isogeny classes that do not contain principally polarized varieties.

MSC:
14K02 Isogeny
14G15 Finite ground fields in algebraic geometry
11G10 Abelian varieties of dimension \(> 1\)
11G25 Varieties over finite and local fields
14K15 Arithmetic ground fields for abelian varieties
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[1] Leonard M. Adleman and Ming-Deh A. Huang, Primality testing and abelian varieties over finite fields, Lecture Notes in Mathematics, vol. 1512, Springer-Verlag, Berlin, 1992. · Zbl 0744.11065
[2] J. V. Armitage, On a theorem of Hecke in number fields and function fields, Invent. Math. 2 (1967), 238 – 246. · Zbl 0143.06304 · doi:10.1007/BF01425516 · doi.org
[3] Pierre Deligne, Variétés abéliennes ordinaires sur un corps fini, Invent. Math. 8 (1969), 238 – 243 (French). · Zbl 0179.26201 · doi:10.1007/BF01406076 · doi.org
[4] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75 – 109. · Zbl 0181.48803
[5] A. Fröhlich, Local fields, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) Thompson, Washington, D.C., 1967, pp. 1 – 41.
[6] Taira Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83 – 95. · Zbl 0203.53302 · doi:10.2969/jmsj/02010083 · doi.org
[7] E. W. Howe, Kernels of polarizations of abelian varieties over finite fields, submitted for publication. · Zbl 0911.11031
[8] P. Deligne, Cristaux ordinaires et coordonnées canoniques, Algebraic surfaces (Orsay, 1976 – 78) Lecture Notes in Math., vol. 868, Springer, Berlin-New York, 1981, pp. 80 – 137 (French). With the collaboration of L. Illusie; With an appendix by Nicholas M. Katz. N. Katz, Serre-Tate local moduli, Algebraic surfaces (Orsay, 1976 – 78) Lecture Notes in Math., vol. 868, Springer, Berlin-New York, 1981, pp. 138 – 202.
[9] Max-Albert Knus, Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 294, Springer-Verlag, Berlin, 1991. With a foreword by I. Bertuccioni. · Zbl 0756.11008
[10] William Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972. · Zbl 0243.14013
[11] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103 – 150.
[12] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. · Zbl 0223.14022
[13] V. B. Mehta and V. Srinivas, Varieties in positive characteristic with trivial tangent bundle, Compositio Math. 64 (1987), no. 2, 191 – 212. With an appendix by Srinivas and M. V. Nori. · Zbl 0639.14024
[14] Frans Oort and Kenji Ueno, Principally polarized abelian varieties of dimension two or three are Jacobian varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 377 – 381. · Zbl 0272.14008
[15] Michael Rosen, Abelian varieties over \?, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 79 – 101.
[16] Hans-Georg Rück, Abelian surfaces and Jacobian varieties over finite fields, Compositio Math. 76 (1990), no. 3, 351 – 366. · Zbl 0742.14037
[17] Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. · Zbl 0584.10010
[18] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. · Zbl 0423.12016
[19] J. Tate, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Séminaire Bourbaki 1968/69, Lecture Notes in Math., vol. 179, Springer-Verlag, Berlin, 1971, exposé 352, pp. 95-110.
[20] William C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. (4) 2 (1969), 521 – 560. · Zbl 0188.53001
[21] André Weil, Zum Beweis des Torellischen Satzes, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa. 1957 (1957), 33 – 53 (German). · Zbl 0079.37002
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