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
![[img]](/style/images/fileicons/application_pdf.png)
| PDF 145Kb |
Abstract
We answer a challenge of Serre by showing that every rational point on the projective curve X + Y
= 17 Z
is of the form (
1,
2, 1) or (
2,
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
81 for which the Fermat quartic X
+ Y
= c Z
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 |
Repository Staff Only: item control page
