Everett W. Howe: Isogeny classes of abelian varieties with no principal polarizations, pp. 203–216 in: Moduli of Abelian Varieties (Carel Faber, Gerard van der Geer, and Frans Oort, eds.), Progr. Math 195, Birkhäuser, Basel, 2001, MR 2002g:11079.
(An unofficial electronic version is 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.)