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Classification of characteristic polynomials of simple supersingular abelian varieties over finite fields. (English) Zbl 1304.14025

By the Honda-Tate theorem, isogeny classes of simple abelian varieties over \({\mathbb F}_q\) correspond to minimum polynomials over \({\mathbb Q}\) of algebraic numbers \(\pi\) all of whose complex embeddings have absolute value \(\sqrt{q}\). The correspondence assigns to an abelian variety \(A\) the characteristic polynomial \(P_A\) of the Frobenius endomorphism. This paper determines, in an easily computable way, the polynomials of this kind that arise from simple supersingular abelian varieties.
One may as well assume that the polynomial has no real roots (otherwise all the roots are all real and the problem becomes trivial). Immediately the problem splits into two cases: if \(q\) is not a square then \(P_A\) is irreducible over \({\mathbb Q}\), and one wants to understand the minimum polynomial over \({\mathbb Q}\) of \(\sqrt{q}\zeta_m\) for some \(m\)th roots of unity \(\zeta_m\). If \(q\) is a square then, as is shown here, \(P_A\) is a power of a cyclotomic polynomial, homogenised with respect to \(\sqrt{q}\).
Both cases are fully worked out here by fairly elementary means. The possible polynomials are listed up to dimension \(7\). A nice consequence is that in certain dimensions no simple supersingular abelian varieties exist: for example, if the dimension of such a variety is prime (and \(>3\)) it must be a Sophie Germain prime and, in particular, congruent to \(5\) mod \(6\).

MSC:

14G15 Finite ground fields in algebraic geometry
11C08 Polynomials in number theory
11G10 Abelian varieties of dimension \(> 1\)
11G15 Complex multiplication and moduli of abelian varieties
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References:

[1] R.P. Brent, On computing factors of cyclotomic polynomials , Mathematics of Computation, pages 131-149, 1993. · Zbl 0785.11011 · doi:10.2307/2152941
[2] M. Deuring, Die typen der multiplikatorenringe elliptischer funktionenkörper , in Abhandlungen aus dem mathematischen Seminar der Universität Hamburg , volume 14, pages 197-272. Springer, 1941. · Zbl 0025.02003 · doi:10.1007/BF02940746
[3] K. Eisenträger, The theorem of Honda and Tate , 2004.
[4] S. Haloui, The characteristic polynomials of abelian varieties of dimensions \(3\) over finite fields , Journal of Number Theory 130 (12) (2010), 2745-2752. · Zbl 1200.14045 · doi:10.1016/j.jnt.2010.06.008
[5] S. Haloui and V. Singh, The characteristic polynomials of abelian varieties of dimension \(4\) over finite fields , in Arithmetic, Geometry, Cryptography and Coding Theory: 13th Conference [on] Arithmetic, Geometry, Cryptography and Coding Theory, CIRM, Marseille, France, March 14-18, 2011: Geocrypt 2011, Bastia, France, June 19-24, 2011 , volume 574, page 59. American Mathematical Soc., 2012. · Zbl 1329.14055 · doi:10.1090/conm/574/11446
[6] D. Maisner and E. Nart, Abelian surfaces over finite fields as Jacobians , Experimental mathematics 11 (3) (2002), 321-337. · Zbl 1101.14056 · doi:10.1080/10586458.2002.10504478
[7] E. Nart and C. Ritzenthaler, Jacobians in isogeny classes of supersingular abelian threefolds in characteristic 2 , Finite Fields and their applications 14 (3) (2008), 676-702. · Zbl 1159.14023 · doi:10.1016/j.ffa.2007.09.006
[8] F. Oort, Subvarieties of moduli spaces , Inventiones mathematicae 24 (2) (1974), 95-119. · Zbl 0259.14011 · doi:10.1007/BF01404301
[9] R.S. Pierce, Associative algebras , Springer Verlag, 1982. · Zbl 0497.16001
[10] K. Rubin and A. Silverberg, Supersingular abelian varieties in cryptology , in Advances in Cryptology - CRYPTO 2002, pages 336-353, Springer, 2002. · Zbl 1026.94540 · doi:10.1007/3-540-45708-9_22
[11] J.P. Serre, Local fields , Springer, 1979.
[12] J. Tate, Endomorphisms of abelian varieties over finite fields , Inventiones mathematicae 2 (2) (1966), 134-144. · Zbl 0147.20303 · doi:10.1007/BF01404549
[13] J. Tate, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda) , Séminaire Bourbaki vol. 1968/69 Exposés 347-363, pages 95-110, 1971. · Zbl 0212.25702
[14] W.C. Waterhouse, Abelian varieties over finite fields , Ann. Sci. Ecole Norm. Sup.(4) 2 (1969), 521-560. · Zbl 0188.53001
[15] C. Xing, On supersingular abelian varieties of dimension two over finite fields , Finite Fields and Their Applications 2 (4) (1996), 407-421. · Zbl 0923.11091 · doi:10.1006/ffta.1996.0024
[16] H.J. Zhu, Supersingular abelian varieties over finite fields , Journal of Number Theory 86 (1) (2001), 61-77. · Zbl 1007.11036 · doi:10.1006/jnth.2000.2562
[17] H.J. Zhu, Group structures of elementary supersingular abelian varieties over finite fields , J. Number Theory 81 (2000), 292-309. · Zbl 1096.11502 · doi:10.1006/jnth.1999.2463
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