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Bounds on polarizations of abelian varieties over finite fields.

Suppose ̂ is an isogeny class of abelian varieties over a finite field. If the varieties in ^ are simple then Theorems 1.1 and 1.2 of [3] give some relatively weak sufficient conditions for there to be a principally polarized variety in <€, perhaps the most striking of these being that the varieties in # be odd-dimensional. However, the examples in Sections 12 and 13 of [2] show that there do exist isogeny classes of abelian varieties over finite fields that do not contain principally polarized varieties. These observations lead to two questions: First, given an arbitrary isogeny class ̂ of abelian varieties over a finite field k, what is the degree of the smallest polarization of a variety in ̂ ? And second, what is the degree of the smallest field extension of k over which the varieties in ̂ become isogenous to a principally polarized variety? In this paper we will provide upper bounds for these numbers in terms of the dimension of the varieties in ̂ and the cardinality of the base field k.