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Integral points on hyperelliptic curves. (English) Zbl 1168.11026

Let \(C\) be a hyperelliptic curve defined by the equation \(Y^2 = f(X)\), where \(f(X)\) is an irreducible polynomial of \({\mathbb Z}[X]\) of degree \(\geq 5\). We denote by \(J\) the Jacobian of \(C\). Suppose that the following assertions are satisfied:
(1) A rational point on \(C\) is known.
(2) A Mordell-Weil basis for \(J( {\mathbb Q})\) is known.
(3) The canonical height \(\hat{h} : J({\mathbb Q}) \rightarrow {\mathbb R} \) is explicitly computable and we have explicit bounds for the difference \[ \mu_1 \leq h(D)-\hat{h}(D) \leq \mu_1^{\prime}, \] where \(h\) is an appropriate normalized logarithmic height on \(J\).
In the paper under review, the authors present a new method for explicitly computing the integral points of \(C\). Using the assumptions (1) and (2), they give a completely explicit upper bound for the integral points of \(C\). Next, under the assumption (3), they show that a combination of this bound with a powerful refinement of the Mordell-Weil sieve, is capable of determining all the integral points of \(C\).
As an illustration of this method the authors determine the integral points of the hyperelliptic curves \(Y^2-Y = X^5-X\) and \({Y\choose 2} ={X\choose 5}\).

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11J86 Linear forms in logarithms; Baker’s method
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