Elliptic curves with \(p\)-Selmer growth for all \(p\) (Q2871290)

Assume throughout the review that the Tate-Shafarevich groups of all elliptic curves are finite. \textit{T. Dokchitser} and \textit{V. Dokchitser} [Acta Arith. 137, No. 2, 193--197 (2009; Zbl 1275.11097)] proved that there exist elliptic curves over number fields such that their rank grows in every quadratic extension. However, this is impossible over \(\mathbb Q\). For a large class of curves, it is also known that the size of their \(2\)-Selmer group will not grow in all quadratic extensions.NEWLINENEWLINEIn the paper under review the author shows that there exist semistable elliptic curves such that their \(2\)-Selmer group grows in every biquadratic extension, and that for any odd prime \(p\), the size of the \(p\)-Selmer group grows in any \(D_{2p}\)-extension, where \(D_{2p}\) is the dihedral group of order \(2p\). Sufficient conditions on the elliptic curve to exhibit this type of behaviour are also given.