Granville, Andrew Rational and integral points on quadratic twists of a given hyperelliptic curve. (English) Zbl 1129.11028 Int. Math. Res. Not. 2007, No. 8, Article ID rnm027, 25 p. (2007). The author considers the set of rational and integral points of twists of a given hyperelliptic curve defined over \(\mathbb{Q}\). The main result of the paper assures that there are few twists which have any rational or integral point, under the hypothesis that the \(abc\) conjecture holds. This is done in the following steps.Denote by \(\mathcal{C} : y^2=f(x)\) an affine equation for the hyperelliptic curve of genus \(g\) with \(f\in\mathbb{Z}[x]\). Its \(d\)-th quadratic twist is given by the equation \(\mathcal{C}_d : dy^2=f(x)\). Supposing the \(abc\) conjecture, the first result is an upper bound for the size of integral (resp. rational) points, if \(g\geq1\), resp. \(g\geq2\), in terms of \(d\) and \(\deg(f)\). As a consequence an upper bound is obtained for the number of square-free integers \(d\) such that \(|d|\leq D\) for which \(\mathcal{C}_d\) has integral or rational points. The bound is expressed in terms of \(D\) and \(\deg(f)\).Next, instead of considering just the set of rational or integral points, the author considers them up to automorphism. It is denoted by \(c_d(\mathbb{Z})\), resp. \(c_d(\mathbb{Q})\), the number of automorphism classes of non trivial integral, resp. rational, points of \(\mathcal{C}_d\). It is conjectured that for \(g\) sufficiently large, there are only finitely many square-free integers \(d\) for which \(c_d(\mathbb{Z})>1\), resp. \(c_d(\mathbb{Q})>2\).In order to quantify this conjecture, the author uses not only the \(abc\) conjecture, but also the Bombieri-Lang conjecture. This latter states the following: given a variety \(X\) of general type defined over a number field \(K\), there exists a proper closed subvariety \(S\subset X\) such that for any number field \(L\) containing \(K\), the set \(X(L)\setminus S\) of \(L\)-rational points of \(X\) lying outside \(S\) is finite. Under this conjecture, and supposing \(g>1\), it is proved that the set of square-free integers \(d\) for which \(c_d(\mathbb{Q})\geq2\) may be parametrized by the rational points on a finite number of curves of genus 0 and 1, together with finitely many exceptional \(d\). Finally the \(abc\) conjecture (resp. the Bombieri-Lang conjecture) is used to quantify the number of square-free integers \(d\leq D\) for which \(c_d(\mathbb{Z})\geq2\) (resp. \(c_d(\mathbb{Q})\geq2\)). Reviewer: Amilcar Pacheco (Rio de Janeiro) Cited in 2 ReviewsCited in 14 Documents MSC: 11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields 14G05 Rational points 11D41 Higher degree equations; Fermat’s equation 14H45 Special algebraic curves and curves of low genus 14H30 Coverings of curves, fundamental group Keywords:rational points; integral points; twists of a given curve × Cite Format Result Cite Review PDF Full Text: DOI