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Mathematics of Computation

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A computational approach to the 2-torsion structure of abelian threefolds

Author: John Cullinan
Journal: Math. Comp. 78 (2009), 1825-1836
MSC (2000): Primary 11G10
Published electronically: January 22, 2009
MathSciNet review: 2501078
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Abstract: Let $ A$ be a three-dimensional abelian variety defined over a number field $ K$. Using techniques of group theory and explicit computations with MAGMA, we show that if $ A$ has an even number of $ \mathbf{F}_{\mathfrak{p}}$-rational points for almost all primes $ \mathfrak{p}$ of $ K$, then there exists a $ K$-isogenous $ A'$ which has an even number of $ K$-rational torsion points. We also show that there exist abelian varieties $ A$ of all dimensions $ \geq 4$ such that $ \char93 A_{\p}(\mathbf{F}_{\mathfrak{p}})$ is even for almost all primes $ \mathfrak{p}$ of $ K$, but there does not exist a $ K$-isogenous $ A'$ such that $ \char93 A'(K)_{tors}$ is even.

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Additional Information

John Cullinan
Affiliation: Department of Mathematics, Bard College, P.O. Box 5000, Annandale-on-Hudson, New York 12504

Keywords: Abelian variety, torsion points
Received by editor(s): February 26, 2007
Received by editor(s) in revised form: August 2, 2008
Published electronically: January 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society