Isogeny classes of abelian varieties with no principal polarizations

Mathematics – Algebraic Geometry

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

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.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Isogeny classes of abelian varieties with no principal polarizations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Isogeny classes of abelian varieties with no principal polarizations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Isogeny classes of abelian varieties with no principal polarizations will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-199091

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.