The Mathematical Institute, University of Oxford, Eprints Archive

Covering collections and a challenge problem of Serre

Flynn, E. V. and Wetherell, J. L. (2001) Covering collections and a challenge problem of Serre. Acta Arithmetica, XCVIII (2). pp. 197-205. ISSN 0065-1036

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Abstract

We answer a challenge of Serre by showing that every rational point on the projective curve X$^4$ + Y$^4$ = 17 Z$^4$ is of the form ($\pm$1, $\pm$2, 1) or ($\pm$2, $\pm$1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curves of high rank. This is the only value of c$\le$81 for which the Fermat quartic X$^4$ + Y$^4$ = c Z$^4$ cannot be solved trivially, either by local considerations or maps to elliptic curves of rank 0, and it seems likely that our approach should give a method of attack for other nontrivial values of c.

Item Type:Article
Uncontrolled Keywords:Fermat Quartics, Covering Techniques.
Subjects:A - C > Algebraic geometry
H - N > Number theory
Research Groups:Number Theory Group
ID Code:257
Deposited By:E. Victor Flynn
Deposited On:12 Jul 2006
Last Modified:20 Jul 2009 14:19

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