Elliptic surfaces and intersections of adelic \(\mathbb{R}\)-divisors (Q6582313)

Let \(k=\overline{\mathbb{Q}}(B)\) be the function field of a smooth projective curve \(B\), and let \(\pi\colon\mathcal{E}\to B\) be a non-isotrivial elliptic surface over \(B\), and therefore also an elliptic curve \(E\) over \(k\). The article under review proves the non-degeneracy of the Arakelov-Zhang pairing as a biquadratic form on \(E(k)\otimes\mathbb{R}\).\N\NThis result has consequences of a startling variety. For example, the authors deduce from their main theorem a Bogomolov-like statement about the specializations of \(\mathcal{E}\) that have lower rank. Specifically, their Theorem 6.1 implies much more than the following result:\N\NTheorem: Let \(\Lambda\subset E(k)\) be a subgroup of rank \(2\) with \(\Lambda/\Lambda_{\mbox{tors}}\) generated by \(S_1,S_2\). There is a constant \(\varepsilon\geq 0\) depending on \(\Lambda\) such that the following set is finite: \[\{t\in B(\overline{\mathbb{Q}}) \mid \hat{h}_t(S_1)+\hat{h}_t(S_2)\leq\varepsilon\}\]\N\NThis is just one example; their Theorem 6.4 lists seven equivalent formulations of their main theorem ranging from families of abelian varieties of high dimension to a \(\mathbb{R}\)-linear dependence criteria in terms of heights on infinitely many fibres of \(\mathcal{E}\).\N\NAnd if that weren't enough, the authors also prove in the appendix an equidistribution result on adelically metrized \(\mathbb{R}\)-divisors on projective varieties over a number field.