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## Covering collections and a challenge problem of Serre

### 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

Topics: Algebraic geometry, Number theory
Year: 2001
OAI identifier: oai:generic.eprints.org:257/core69

### Citations

1. (1995). 2-descent on the jacobians of hyperelliptic curves.
2. (1997). A Flexible Method for Applying Chabauty’s Theorem.
3. (1997). Bounding the Number of Rational Points on Certain Curves of High Rank, PhD Dissertation
4. (1999). Chabauty methods and covering techniques applied to generalised Fermat equations. PhD Dissertation,
5. (1998). Computing a Selmer group of a Jacobian using functions on the curve.
6. Computing the p-Selmer group of an elliptic curve.
7. (1997). Cycles of quadratic polynomials and rational points on a genus-two curve.
8. (1999). Finding Rational Points on Bielliptic Genus 2 Curves.
9. (2000). Implementing 2-Descent for Jacobians of Hyperelliptic Curves.
10. (1989). Lectures on the Mordell-Weil Theorem, Transl. and ed. by Martin Brown. From notes by Michel Waldschmidt.
11. (1989). On Heterogeneous Spaces,
12. (1996). Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2.
13. (1968). Rational Points on a Class of Algebraic Curves,
14. (1987). Rational Points on Certain Families
15. (1954). Remarques sur un me´moire d’Hermite,
16. (1941). Sur les points rationnels des courbes alge´briques de genre supe´rieur a` l’unite´. Comptes Rendus,
17. (1986). The Arithmetic of Elliptic Curves.
18. (1969). The p-torsion of Elliptic Curves is Uniformly Bounded,

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