Isogeny Classes of Abelian Varieties with no Principal Polarizations

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 t 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 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.