Integral points on hyperelliptic curves

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 $J\left(\mathbb{Q}\right)$. 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 $\left(\genfrac{}{}{0.0pt}{}{Y}{2}\right)=\left(\genfrac{}{}{0.0pt}{}{X}{5}\right)$.