(An uncorrected official version and a corrected unofficial version are available.)

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 extension of k with a certain non-abelian Galois group. (Silverberg and Zarhin have a new construction that works over an arbitrary number field.)

NOTE: The official published version of this paper contains an error. Theorem 3.2 is incorrect. The version of the paper on the arXiv (the “corrected unofficial version” linked to above) points out the error. Unfortunately, there seems to be no way to fix the problem. The construction in the first part of the paper still works; the problem occurs in the “more general framework” mentioned above. Thanks to Armand Brumer for pointing this out to me.