Open Access
2013 Explicit Chabauty over number fields
Samir Siksek
Algebra Number Theory 7(4): 765-793 (2013). DOI: 10.2140/ant.2013.7.765


Let C be a smooth projective absolutely irreducible curve of genus g2 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 KC(K) is in fact equal to C(K). This criterion is likely to be successful if rd(g1). 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.


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Samir Siksek. "Explicit Chabauty over number fields." Algebra Number Theory 7 (4) 765 - 793, 2013.


Received: 6 July 2010; Revised: 23 July 2012; Accepted: 31 October 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1330.11043
MathSciNet: MR3095226
Digital Object Identifier: 10.2140/ant.2013.7.765

Primary: 11G30
Secondary: 14C20 , 14K20

Keywords: abelian integral , Chabauty , Coleman , divisor , generalized Fermat , Jacobian , Mordell–Weil sieve , rational points

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 4 • 2013
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