## 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 $$\ell$$.

### MSC:

 11G10 Abelian varieties of dimension $$> 1$$ 14G15 Finite ground fields in algebraic geometry 14K10 Algebraic moduli of abelian varieties, classification
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### References:

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