Completely faithful Selmer groups over Kummer extensions
We study the Selmer groups of elliptic curves over Galois extensions of number fields whose Galois group is isomorphic to the semidirect product of two copies of the -adic numbers . In particular, we give examples where its Pontryagin dual is a faithful torsion module under the Iwasawa algebra of . 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.