an:00825439
Zbl 0832.14033
Howe, Everett W.
Bounds on polarizations of abelian varieties over finite fields
EN
J. Reine Angew. Math. 467, 149-155 (1995).
0075-4102 1435-5345
1995
j
14K02 14K15 14G15
isogenous abelian variety; abelian variety over a finite field; polarization; principally polarized variety
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.
E.W.Howe (Ann Arbor)