#### Vol. 2, No. 8, 2008

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Integral points on hyperelliptic curves

### Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll and Szabolcs Tengely

Vol. 2 (2008), No. 8, 859–885
##### Abstract

Let $C:{Y}^{2}={a}_{n}{X}^{n}+\cdots +{a}_{0}$ be a hyperelliptic curve with the ${a}_{i}$ rational integers, $n\ge 5$, 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 $J\left(ℚ\right)$. 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 ${Y}^{2}-Y={X}^{5}-X$ and $\left(\genfrac{}{}{0.0pt}{}{Y}{2}\right)=\left(\genfrac{}{}{0.0pt}{}{X}{5}\right)$.

##### Keywords
curve, integral point, Jacobian, height, Mordell–Weil group, Baker's bound, Mordell–Weil sieve
Primary: 11G30
Secondary: 11J86