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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A computational approach to the 2-torsion structure of abelian threefolds
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by John Cullinan PDF
Math. Comp. 78 (2009), 1825-1836 Request permission

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 $\#A_{\mathbb {p} }(\mathbf {F}_{\mathfrak {p}})$ is even for almost all primes $\mathfrak {p}$ of $K$, but there does not exist a $K$-isogenous $A’$ such that $\# A’(K)_{tors}$ 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
  • Email: cullinan@bard.edu
  • Received by editor(s): February 26, 2007
  • Received by editor(s) in revised form: August 2, 2008
  • Published electronically: January 22, 2009
  • © Copyright 2009 American Mathematical Society
  • Journal: Math. Comp. 78 (2009), 1825-1836
  • MSC (2000): Primary 11G10
  • DOI: https://doi.org/10.1090/S0025-5718-09-02218-2
  • MathSciNet review: 2501078