#### DOCUMENTA MATHEMATICA,
Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 443-478

** Y. Hachimori and O. Venjakob **
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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