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: BirkhĂ¤user (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].

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].