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 \(6\).


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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