Completely faithful Selmer groups over Kummer extensions (Q1419617)

Summary: We study the Selmer groups of elliptic curves over Galois extensions of number fields whose Galois group \(G\) is isomorphic to the semidirect product of two copies of the \(p\)-adic numbers \(\mathbb{Z}_p\). In particular, we give examples where its Pontryagin dual is a faithful torsion module under the Iwasawa algebra of \(G\). Then we calculate its Euler characteristic and give a criterion for the Selmer group being trivial. Furthermore, we describe a new asymptotic bound of the rank of the Mordell-Weil group in these towers of number fields.