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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