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Computing the $p$-Selmer group of an elliptic curve


Authors: Z. Djabri, Edward F. Schaefer and N. P. Smart
Journal: Trans. Amer. Math. Soc. 352 (2000), 5583-5597
MSC (2000): Primary 11G05, 11Y99; Secondary 14H52, 14Q05
Published electronically: August 21, 2000
MathSciNet review: 1694286
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Abstract:

In this paper we explain how to bound the $p$-Selmer group of an elliptic curve over a number field $K$. Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number fields of degree at most $p^2-1$. Our method is practical for $p=3$, but for larger values of $p$it becomes impractical with current computing power. In the examples we have calculated, our method produces exactly the $p$-Selmer group of the curve, and so one can use the method to find the Mordell-Weil rank of the curve when the usual method of $2$-descent fails.


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Additional Information

Z. Djabri
Affiliation: Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent, CT2 7NF, United Kingdom
Address at time of publication: Riskcare, Piercy House, 7 Copthall Avenue, London EC2R 7NJ, United Kingdom
Email: zmd1@ukc.ac.uk, zdjabri@riskcare.com

Edward F. Schaefer
Affiliation: Department of Mathematics, Santa Clara University, Santa Clara, California 95053
Email: eschaefe@math.scu.edu

N. P. Smart
Affiliation: Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol, BS12 6QZ, United Kingdom
Address at time of publication: Computer Science Department, Woodland Road, University of Bristol, Bristol, BS8 1UB, United Kingdom
Email: nsma@hplb.hpl.hp.com, nigel@cs.bris.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02535-6
Keywords: Elliptic curves, Mordell-Weil rank, Selmer group
Received by editor(s): October 28, 1998
Received by editor(s) in revised form: February 26, 1999, and March 17, 1999
Published electronically: August 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society