Abstract
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 t 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 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Everett W. Howe, Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc. 347 (1995), 2361–2401.
Everett W. Howe, Kernels of polarizations of abelian varieties over finite fields, J. Algebraic Geom. 5 (1996), 583–608.
David Mumford, Abelian. varieties, Tata Inst. Fund. Res. Stud. Math. 5, Oxford University Press, Oxford, 1985.
V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure Appl. Math. 139, Academic Press, San Diego, California, 1994.
I. Reiner, Maximal Orders, London Math. Soc. Monographs 5, Academic Press, London, 1975.
W. Scharlau, Quadratic and Hermitian Forms, Grundlehren Math. Wiss. 270, SpringerVerlag, Berlin, 1985.
Jean-Pierre Serre, Cohomologie galoisienne (Cinguieme edition, revisee et completee), Lecture Notes in Math. 5, Springer Verlag, Berlin, 1994.
A. Silverberg and Yu. G. Zarhin, Self-dual tadic representations of finite groups, preprint (available athttp://www.math.uiuc.edu/Algebraic-Number-Theory/0151/index.html) (1998).
A. Silverberg and Yu. G. Zarhin, Polarizations on abelian varieties and self-dual tactic representations of inertia groups, preprint (available athttp://www.math.uiuc.edu/Algebraic-Number-Theory/0171/index.html) (1999).
A. Silverberg and Yu. G. Zarhin, Polarizations on abelian varieties, preprint (available athttp://www.lanl.gov/abs/math.AG/0002253) (2000).
A. Weil, Adeles and algebraic groups, Progr. Math. 23, Birkhauser, Boston, 1982.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this chapter
Cite this chapter
Howe, E.W. (2001). Isogeny Classes of Abelian Varieties with no Principal Polarizations. In: Faber, C., van der Geer, G., Oort, F. (eds) Moduli of Abelian Varieties. Progress in Mathematics, vol 195. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8303-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8303-0_7
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9509-5
Online ISBN: 978-3-0348-8303-0
eBook Packages: Springer Book Archive