The Mathematical Institute, University of Oxford, Eprints Archive

Application of covering techniques to families of curves

Flynn, E. V. and Redmond, J. (2003) Application of covering techniques to families of curves. Journal of Number Theory, 101 . pp. 376-397. ISSN 0022-314X

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Abstract

Much success in finding rational points on curves has been obtained by using Chabauty's Theorem, which applies when the genus of a curve is greater than the rank of the Mordell-Weil group of the Jacobian. When Chabauty's Theorem does not directly apply to a curve C, a recent modification has been to cover the rational points on C by those on a covering collection of curves D$_i$, obtained by pullbacks along an isogeny to the Jacobian; one then hopes that Chabauty's Theorem applies to each D$_i$. So far, this latter technique has been applied to isolated examples. We apply, for the first time, certain covering techniques to infinite families of curves. We find an infinite family of curves to which Chabauty's Theorem is not applicable, but which can be solved using bielliptic covers, and other infinite families of curves which even resist solution by bielliptic covers. A fringe benefit is an infinite family of Abelian surfaces with non-trivial elements of the Tate-Shafarevich group killed by a bielliptic isogeny.

Item Type:Article
Uncontrolled Keywords:Coverings of Curves, Descent.
Subjects:A - C > Algebraic geometry
H - N > Number theory
Research Groups:Number Theory Group
ID Code:255
Deposited By:E. Victor Flynn
Deposited On:12 Jul 2006
Last Modified:20 Jul 2009 14:19

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