Completely Faithful Selmer Groups over Kummer Extensions

In this paper we study the Selmer groups of elliptic curves over Galois extensions of number fields whose Galois group $G\cong\zp\rtimes\zp$ is isomorphic to the semidirect product of two copies of the $p$-adic numbers $\zp.$ 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.

2000 Mathematics Subject Classification: Primary 11G05, 14K15; Secondary 16S34, 16E65.

Keywords and Phrases: Selmer groups, elliptic curves, Euler characteristics, $p$-adic analytic groups.

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