## Algebra & Number Theory

### Integral points on hyperelliptic curves

#### Abstract

Let $C:Y2=anXn+⋯+a0$ be a hyperelliptic curve with the $ai$ rational integers, $n≥5$, and the polynomial on the right-hand side irreducible. Let $J$ be its Jacobian. We give a completely explicit upper bound for the integral points on the model $C$, provided we know at least one rational point on $C$ and a Mordell–Weil basis for $J(ℚ)$. We also explain a powerful refinement of the Mordell–Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on the genus $2$ hyperelliptic models $Y2−Y=X5−X$ and $Y2=X5$.

#### Article information

Source
Algebra Number Theory, Volume 2, Number 8 (2008), 859-885.

Dates
Revised: 2 September 2008
Accepted: 12 September 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797332

Digital Object Identifier
doi:10.2140/ant.2008.2.859

Mathematical Reviews number (MathSciNet)
MR2457355

Zentralblatt MATH identifier
1168.11026

#### Citation

Bugeaud, Yann; Mignotte, Maurice; Siksek, Samir; Stoll, Michael; Tengely, Szabolcs. Integral points on hyperelliptic curves. Algebra Number Theory 2 (2008), no. 8, 859--885. doi:10.2140/ant.2008.2.859. https://projecteuclid.org/euclid.ant/1513797332

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