Bruin, N. and Flynn, E. V. (2005) Towers of 2covers of hyperelliptic curves. Transactions of the American Mathematical Society, 357 . pp. 43294347. ISSN 00029947

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Abstract
In this article, we give a way of constructing an unramified Galois cover of a hyperelliptic curve. The geometric Galoisgroup is an elementary abelian 2group. 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 multiplicationby2 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 2covers 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:  29 May 2015 18:18 
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