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Mathematics of Computation

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Two-cover descent on hyperelliptic curves

Authors: Nils Bruin and Michael Stoll
Journal: Math. Comp. 78 (2009), 2347-2370
MSC (2000): Primary 11G30; Secondary 14H40.
Published electronically: March 11, 2009
MathSciNet review: 2521292
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Abstract | References | Similar Articles | Additional Information

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
MR Author ID: 653028

Michael Stoll
Affiliation: Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

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
Published electronically: March 11, 2009
Additional Notes: The research of the first author was supported by NSERC
Article copyright: © Copyright 2009 American Mathematical Society