Algebra & Number Theory

Explicit Chabauty over number fields

Samir Siksek

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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.

Article information

Algebra Number Theory, Volume 7, Number 4 (2013), 765-793.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25]
Secondary: 14K20: Analytic theory; abelian integrals and differentials 14C20: Divisors, linear systems, invertible sheaves

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


Siksek, Samir. Explicit Chabauty over number fields. Algebra Number Theory 7 (2013), no. 4, 765--793. doi:10.2140/ant.2013.7.765.

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