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Bounds on polarizations of abelian varieties over finite fields. (English) Zbl 0832.14033
Let $$A$$ be an $$n$$-dimensional abelian variety over a finite field $$k$$ of $$q$$ elements and let $$\Delta = (4n)^n q^{n(n - 1)/2} \min (1/4, (q/4)^{n/2})$$. The main result of this paper is that $$A$$ is isogenous over $$k$$ to an abelian variety that has a polarization of degree at most $$\Delta$$. Furthermore, there is a field extension $$\ell$$ of $$k$$ of degree at most $$1 + \Delta^{1/2}$$ over $$k$$ such that $$A$$ becomes isogenous to a principally polarized variety when the base field is extended to $$\ell$$. If the generalized Riemann hypothesis is true and if the variety $$A$$ is simple and has dimension larger than one, the number $$\Delta$$ can be replaced by $$144 \log^4 \Delta$$ in the preceding statements.
##### MSC:
 14K02 Isogeny 14K15 Arithmetic ground fields for abelian varieties 14G15 Finite ground fields in algebraic geometry
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