We provide a simple method of constructing isogeny classes of abelian
varieties over certain fields k such that no variety in the isogeny
class has a principal polarization. In particular, given a field k,
a Galois extension l of k of odd prime degree p, and an
elliptic curve E over k that has no complex multiplication
over k and that has no k-defined p-isogenies to another
elliptic curve, we construct a simple (p-1)-dimensional abelian
variety X over k such that every polarization of every abelian
variety isogenous to X has degree divisible by p2.
We note that for every odd prime p and every number field k,
there exist l and E as above. We also provide a general
framework for determining which finite group schemes occur as kernels of
polarizations of abelian varieties in a given isogeny class.
Our construction was inspired by a similar construction of
Silverberg and Zarhin; their construction requires that the base
field k have positive characteristic and that there be a Galois
of k with a certain non-abelian Galois group.
This paper will appear in the volume Moduli of Abelian Varieties
(Texel Island 1999).
Everett W. Howe <email@example.com>