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Proceedings of the London Mathematical Society Advance Access originally published online on November 12, 2007
Proceedings of the London Mathematical Society 2008 96(2):312-334; doi:10.1112/plms/pdm049
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© 2007 London Mathematical Society

Isogeny classes of abelian varieties over function fields

Yuri G. Zarhin

Department of Mathematics
Pennsylvania State University
University Park, PA 16802
USA

Received 4 July 2006. Revision received 22 May 2007.

We study finiteness problems for isogeny classes of abelian varieties over an algebraic function field K in one variable over the field of complex numbers. In particular, we construct explicitly a non-isotrivial absolutely simple abelian fourfold X over a certain K such that the isogeny class of XxX contains infinitely many mutually non-isomorphic principally polarized abelian varieties. (Such examples do not exist when the ground field is finitely generated over its prime subfield.)

2000 Mathematics Subject Classification 14K02 (primary), 14D07 (secondary).


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