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Two-cover descent on hyperelliptic curves

Author(s): Nils Bruin; Michael Stoll.
Journal: Math. Comp. 78 (2009), 2347-2370.
MSC (2000): Primary 11G30; Secondary 14H40.
Posted: March 11, 2009
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Abstract: We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability of hyperelliptic curves. We also discuss applications of this algorithm to curves of genus $ 1$ and to curves with rational points.

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Additional Information:

Nils Bruin
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
Email: nbruin@sfu.ca

Michael Stoll
Affiliation: Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
Email: Michael.Stoll@uni-bayreuth.de

DOI: 10.1090/S0025-5718-09-02255-8
PII: S 0025-5718(09)02255-8
Keywords: Local-global obstruction, rational points, hyperelliptic curves, descent
Received by editor(s): March 31, 2008
Received by editor(s) in revised form: October 21, 2008
Posted: March 11, 2009
Additional Notes: The research of the first author was supported by NSERC
Copyright of article: Copyright 2009, American Mathematical Society

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