Vol. 7, No. 4, 2013

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Explicit Chabauty over number fields

Samir Siksek

Vol. 7 (2013), No. 4, 765–793
Abstract

Let C be a smooth projective absolutely irreducible curve of genus g 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 C(K) is in fact equal to C(K). This criterion is likely to be successful if r d(g 1). We also show that the only solution to the equation x2 + y3 = z10 in coprime nonzero integers is (x,y,z) = (±3,2,±1). This is achieved by reducing the problem to the determination of K-rational points on several genus-2 curves where K = or (23) 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/