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

HTML articles powered by AMS MathViewer

- 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 $Aβ$ 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 $Aβ$ such that $\# Aβ(K)_{tors}$ is even.## References

- M. Aschbacher,
*Finite group theory*, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 2000. MR**1777008**, DOI 10.1017/CBO9781139175319 - Gregory Butler and John McKay,
*The transitive groups of degree up to eleven*, Comm. Algebra**11**(1983), no.Β 8, 863β911. MR**695893**, DOI 10.1080/00927878308822884 - J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
*$\Bbb {ATLAS}$ of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219** - John Cullinan,
*Local-global properties of torsion points on three-dimensional abelian varieties*, J. Algebra**311**(2007), no.Β 2, 736β774. MR**2314732**, DOI 10.1016/j.jalgebra.2007.02.024 - Marc Hindry and Joseph H. Silverman,
*Diophantine geometry*, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR**1745599**, DOI 10.1007/978-1-4612-1210-2 - Christoph Jansen, Klaus Lux, Richard Parker, and Robert Wilson,
*An atlas of Brauer characters*, London Mathematical Society Monographs. New Series, vol. 11, The Clarendon Press, Oxford University Press, New York, 1995. Appendix 2 by T. Breuer and S. Norton; Oxford Science Publications. MR**1367961** - Nicholas M. Katz,
*Galois properties of torsion points on abelian varieties*, Invent. Math.**62**(1981), no.Β 3, 481β502. MR**604840**, DOI 10.1007/BF01394256 - Peter Kleidman and Martin Liebeck,
*The subgroup structure of the finite classical groups*, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR**1057341**, DOI 10.1017/CBO9780511629235 - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR**553598** - J-P. Serre. Letter to J. Cullinan, 2006.

## 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