A Note on the Iwasawa λ-Invariants of Real Abelian Number Fields

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Let k be a real abelian number field with Galois group Δ and p an odd prime number. Assume that the order of Δ is not divisible by p. Let Ψ be an irreducible Qp-character of Δ. Denote by λp(Ψ ) the Ψ-component of the λ-invariant associated to the cyclotomic Zp-extension of k. Then Greenberg conjecture for the Ψ-components states that λp(Ψ ) is always zero for any Ψ and p. Although some efficient criteria for the conjecture to be true are given, very little is known about it except for k =Q or the trivial character case. There is another λ-invariant. Denote by λp*(Ψ ) the λ-invariant associated to the p -adic L-function related to Ψ. One can know λp*(Ψ ) by computing the Iwasawa power series attached to Ψ. The Iwasawa main conjecture proved by Mazur and Wiles says that the inequality λp(Ψ ) ≤ λp*(Ψ ) holds. In this paper, we give a necessary and sufficient condition for this inequality to be strict in terms of special values of p -adic L-functions. This result enables us to obtain a criterion for Greenberg’s conjecture for Ψ-components to be true when the corresponding Iwasawa power series is irreducible.
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