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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 \(m=\# A({\mathbb F}_q)\).

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