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Isogeny classes of abelian varieties with no principal polarizations. (English) Zbl 1079.14531
Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser (ISBN 3-7643-6517-X/hbk). Prog. Math. 195, 203-216 (2001).
Summary: 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 A. Silverberg and Yu. G. Zarhin [Math. Proc. Camb. Philos. Soc. 133, No. 2, 223–233 (2002; Zbl 1011.14011)]; 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.
For the entire collection see [Zbl 0958.00023].

MSC:
14K02 Isogeny
11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
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