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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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