Computing Mordell-Weil ranks of cyclic covers of elliptic surfaces (Q2718945)

Let \(\pi:E\to\mathbb P^1\) be a smooth complex relatively minimal elliptic surface with a section. For each integer \(r\geq 1\), let \(\pi_r:E_r\to\mathbb P^1\) be the relatively minimal compactification of the Néron model of the generic fiber of \(E\times_{\mathbb P^1}\mathbb P^1\) of the pull-back of \(E\) by the morphism of \(\mathbb P^1\) defined by \(t\to t^r\). \textit{L. A. Fastenberg} [Duke Math. J. 89, 217-224 (1997; Zbl 0903.14006)] proved that under certain conditions on \(E\) the rank of the Mordell-Weil group of sections of \(E_r\) is bounded independently of \(r\). In the paper under review she uses this result to compute or give explicit upper bounds for the rank of \(E_r(\mathbb P^1)\) for several examples.