Two-cover descent on hyperelliptic curves

Bruin, Nils; Stoll, Michael

doi:10.1090/S0025-5718-09-02255-8

Two-cover descent on hyperelliptic curves
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by Nils Bruin and Michael Stoll;

Math. Comp. 78 (2009), 2347-2370

DOI: https://doi.org/10.1090/S0025-5718-09-02255-8

Published electronically: 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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