Algebra & Number Theory

Explicit Chabauty over number fields

Samir Siksek

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729980

Digital Object Identifier
doi:10.2140/ant.2013.7.765

Mathematical Reviews number (MathSciNet)
MR3095226

Zentralblatt MATH identifier
1330.11043

Subjects
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

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

Citation

Siksek, Samir. Explicit Chabauty over number fields. Algebra Number Theory 7 (2013), no. 4, 765--793. doi:10.2140/ant.2013.7.765. https://projecteuclid.org/euclid.ant/1513729980


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