Real polynomials with all roots on the unit circle and abelian varieties over finite fields.(English)Zbl 0931.11023

J. Number Theory 73, No. 2, 426-450 (1998); corrigendum ibid. 83, 182 (2000).
The Honda-Tate theorem associates to an isogeny class of an abelian variety $$A$$ over $${\mathbb F}_q$$ its characteristic polynomial of Frobenius $$f_A\in {\mathbb Z}[x]$$. The Weil “conjectures” imply that all the complex roots of the monic $$f_A$$ of degree $$2\cdot \dim A$$ have absolute value $$q^{1/2}$$ and moreover $$f_A=(x^{2n}+q^n)+a_1(x^{2n-1}+q^{n-1}x)+\cdots +a_{n-1}(x^{n+1}+qx^{n-1})+a_nx^n$$ with $$a_1,\dots ,a_n\in {\mathbb Z}$$. The abelian variety is ordinary if and only if $$a_n$$ is coprime to $$q$$.
In this paper one counts abelian varieties as above (ordinary or not) by computing the volume of $$V_n\subset {\mathbb R}^n$$ defined by $$(b_1,\dots ,b_n)\in V_n$$ if the polynomial $$(x^{2n}+1)+b_1(x^{2n-1}+x)+\cdots +b_{n-1}(x^{n+1}+x^{n-1}) +b_nx^n$$ has all its complex roots on the unit circle and if $$1$$ or $$-1$$ are roots then their multiplicity is even. The classes that one counts are seen as lattice points in $$V_n$$. In this way one finds estimates for the number of isogeny classes of abelian varieties $$A$$ of dimension $$n$$ over $${\mathbb F}_q$$. A further result is that for a suitable range of integers $$m$$ there is an abelian variety $$A$$ over $${\mathbb F}_q$$ of dimension $$n$$ with $$m=\# A({\mathbb F}_q)$$.

MSC:

 11G25 Varieties over finite and local fields 14G20 Local ground fields in algebraic geometry 11R09 Polynomials (irreducibility, etc.)
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