Non-thin rank jumps for double elliptic \(K3\) surfaces (Q6636592)

Let \(X\) be a \(K3\) surface endowed with a structure of Jacobian elliptic surface over a number field \(K\) with a non-isotrivial and without non-reduced fibres \(\pi:X\to\mathbb{P}_K^1\).\N\NThe aim of the article under review is to study the non-thin-ness of the set \(\mathscr{R}(X,\,\pi,\,K)\) of \(K\)-rational points \(t\) of \(\mathbb{P}_K^1\) such that the rank of \(K\)-rational points on the fibre \(X_t:=\pi^{-1}(t)\) over \(t\) is strictly larger than the rank of the Mordell-Weil lattice \(\mathrm{MW}(X,\,\pi)\) associated to the fibration \(\pi\).\N\NThe main theorem states that: If there exists a different elliptic fibration \(\nu:X\to\mathbb{P}_K^1\), then, they give two equivalent conditions for the non-thin-ness of the set \(\mathscr{R}(X,\,\pi)\) in \(\mathbb{P}_K^1\); that the set \(X(K)\) of \(K\)-rational points in \(X\) is Zariski dense; that the fibration \(\pi\) has the rank jump property, that is, the set \(\mathscr{R}(X,\,\pi,\,K)\) is an infinite set.\N\NIn particular, in the case where \(X\) is the Kummer surface of product type \(\mathrm{Km}(E\times F)\) of elliptic curves \(E\) and \(F\), they give three equivalent conditions for the non-thin-ness of the set \(\mathscr{R}(X,\,\pi)\) in \(\mathbb{P}_K^1\). Besides, they conclude with criteria for these equivalent conditions to hold in terms of the elliptic curves \(E\) and \(F\), and of the Mordell-Weil lattice of \(\pi\).\N\NThey give an innovative new criterion involving Salient curves for the linear-independence of multisections. The assumption of the existence of a different elliptic fibration \(\nu\) is still needed to get an appropriate Salient curve as in Lemmas 3.3 and 4.2 that play crucial parts in the proof of the main theorem.