The Mathematical Institute, University of Oxford, Eprints Archive

Towers of 2-covers of hyperelliptic curves

Bruin, N. and Flynn, E. V. (2005) Towers of 2-covers of hyperelliptic curves. Transactions of the American Mathematical Society, 357 . pp. 4329-4347. ISSN 0002-9947

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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. Especially 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 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.

Item Type:Article
Uncontrolled Keywords:Covers of Curves, Hyperelliptic Curves, Rational Points, Descent, Method of Chabauty.
Subjects:A - C > Algebraic geometry
H - N > Number theory
Research Groups:Number Theory Group
ID Code:250
Deposited By:E. Victor Flynn
Deposited On:12 Jul 2006
Last Modified:20 Jul 2009 14:19

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