### Isogeny classes of abelian varieties with no principal polarizations, by Everett W. Howe

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 *p*^{2}.
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
extension
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 <however@alumni.caltech.edu>