Explicit Chabauty over number fields

Siksek, Samir

doi:10.2140/ant.2013.7.765

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Explicit Chabauty over number fields

Samir Siksek

Vol. 7 (2013), No. 4, 765–793

DOI: 10.2140/ant.2013.7.765

Abstract

Let $C$ be a smooth projective absolutely irreducible curve of genus $g \geq 2$ over a number field $K$ of degree $d$ , and let $J$ denote its Jacobian. Let $r$ denote the Mordell–Weil rank of $J (K)$ . We give an explicit and practical Chabauty-style criterion for showing that a given subset $K \subseteq C (K)$ is in fact equal to $C (K)$ . This criterion is likely to be successful if $r \leq d (g - 1)$ . We also show that the only solution to the equation $x^{2} + y^{3} = z^{10}$ in coprime nonzero integers is $(x, y, z) = (\pm 3, - 2, \pm 1)$ . This is achieved by reducing the problem to the determination of $K$ -rational points on several genus- $2$ curves where $K = ℚ$ or $ℚ (\sqrt[3]{2})$ and applying the method of this paper.

Keywords

Chabauty, Coleman, jacobian, divisor, abelian integral, Mordell–Weil sieve, generalized Fermat, rational points

Mathematical Subject Classification 2010

Primary: 11G30

Secondary: 14K20, 14C20

Milestones

Received: 6 July 2010

Revised: 23 July 2012

Accepted: 31 October 2012

Published: 29 August 2013

Authors

Samir Siksek
Department of Mathematics University of Warwick Coventry CV4 7AL United Kingdom
http://www.warwick.ac.uk/~maseap/

Vol. 7, No. 4, 2013