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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on good reduction of simple Abelian varieties


Author: C. Adimoolam
Journal: Proc. Amer. Math. Soc. 64 (1977), 196-198
MSC: Primary 14K15; Secondary 14G25
DOI: https://doi.org/10.1090/S0002-9939-1977-0447259-X
MathSciNet review: 0447259
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Abstract: In this note it is shown that the reduction of a simple abelian variety of dimension $ \geqslant 2$, defined over an algebraic number field, at any finite good prime need not be simple. We give an example of a two-dimensional simple abelian variety defined over an algebraic number field whose reduction at any finite prime is isogenous either to a product of ordinary elliptic curves or to a product of supersingular elliptic curves.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0447259-X
Keywords: Simple abelian variety, good reduction, p-rank of an abelian variety, Barsotti-Tate group, Dieudonné module, indefinite quaternion algebra, ordinary and supersingular elliptic curves
Article copyright: © Copyright 1977 American Mathematical Society