Abstract
Let ℓ be a regular odd prime number, k the ℓth cyclotomic field, k∞ the cyclotomic ℤℓ-extension of k, K a cyclic extension of k of degree ℓ, and = K · k∞. Under the assumption that there are exactly three places not over ℓ that ramify in the extension K∞/k∞ and K satisfies some additional conditions, we study the structure of the Iwasawa module Tℓ(K∞) of K∞ as a Galois module. In particular, we prove that Tℓ(K∞) is a cyclic G(K∞/k∞)-module and the Galois group Γ = G(K∞/K) acts on Tℓ(K∞) as \(\sqrt \chi \), where \(\chi :\Gamma \to \mathbb{Z}_\ell^ \times \) is the cyclotomic character.
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This article was submitted by the author simultaneously in Russian and English
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 307, pp. 78–99.
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Kuz’min, L.V. Arithmetic of Certain ℓ-Extensions Ramified at Three Places. Proc. Steklov Inst. Math. 307, 65–84 (2019). https://doi.org/10.1134/S008154381906004X
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DOI: https://doi.org/10.1134/S008154381906004X