Mathematics – Algebraic Geometry
Scientific paper
2000-02-28
pp. 203--216 in: Moduli of Abelian Varieties (Carel Faber, Gerard van der Geer, and Frans Oort, eds.), Progr. Math 195, Birkh\
Mathematics
Algebraic Geometry
13 pages, AMS-TeX, with updated references. To appear in the volume "Moduli of Abelian Varieties (Texel Island 1999)"
Scientific paper
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.
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