Abelian varieties over finite fields with a specified characteristic polynomial modulo \(\ell\). (English) Zbl 1156.11322

From the introduction: Let \(\mathbb F\) be a finite field of characteristic \(p\) and order \(q\), and \(\ell\) a prime not equal to \(p\). Let \[ P(T)= (T^{2g} + q^g) + a_1(T^{2g-1} + q^{g-1}T) + \dots+ a_{g-1}(T^{g+1} + qT^{g-1}) + a_gT^g \]
be a polynomial. The goal of this paper is to estimate the number of isogeny classes of abelian varieties over \(\mathbb F\) of dimension \(g\) for which the characteristic polynomial for the action of Frobenius is congruent to \(P(T)\) modulo \(\ell\).
As an application we find the proportion of isogeny classes of abelian varieties with a rational point of order \(\ell\).


11G10 Abelian varieties of dimension \(> 1\)
14G15 Finite ground fields in algebraic geometry
14K10 Algebraic moduli of abelian varieties, classification
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