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Towers of 2-covers of hyperelliptic curves

Authors: Nils Bruin and E. Victor Flynn
Journal: Trans. Amer. Math. Soc. 357 (2005), 4329-4347
MSC (2000): Primary 11G30; Secondary 11G10, 14H40
Published electronically: June 22, 2005
MathSciNet review: 2156713
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Abstract: In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian $2$-group. The construction does not make use of the embedding of the curve in its Jacobian, and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by-$2$ map of an embedding of the curve in its Jacobian.

We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. In particular the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves.

As an application, we determine the rational points on the genus $2$ curve arising from the question of whether the sum of the first $n$ fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds.

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Additional Information

Nils Bruin
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

E. Victor Flynn
Affiliation: Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom

Keywords: Covers of curves, hyperelliptic curves, rational points, descent, method of Chabauty
Received by editor(s): July 9, 2001
Received by editor(s) in revised form: September 22, 2002
Published electronically: June 22, 2005
Additional Notes: The first author was supported by the Pacific Institute for the Mathematical Sciences, Simon Fraser University and the University of British Columbia
The second author received financial support from EPSRC Grant Number GR/R82975/01
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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