## Algebra & Number Theory

### Explicit Chabauty over number fields

Samir Siksek

#### 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 $K=ℚ$ or $ℚ(23)$ and applying the method of this paper.

#### Article information

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

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

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

#### 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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