Kummer generators and lambda invariants

TL;DR

This paper investigates conditions for the Iwasawa lambda invariant to be at least 2 in certain imaginary quadratic fields, constructing units that serve as Kummer generators and linking algebraic properties with 3-adic L-functions.

Contribution

It introduces a method to construct units in the first level of the Z_3-extension that relate to the lambda invariant, providing an algebraic interpretation of the condition mbda  2.

Findings

Constructed units potentially serving as Kummer generators for unramified extensions.

Provided algebraic interpretation of mbda  2 in terms of these units.

Connected conditions for mbda  2 with 3-adic L-functions and class numbers.

Abstract

Let $F_0=\mathbf Q(\sqrt{-d})$ be an imaginary quadratic field with $3\nmid d$ and let $K_0=\mathbf Q(\sqrt{3d})$. Let $\varepsilon_0$ be the fundamental unit of $K_0$ and let $\lambda$ be the Iwasawa $\lambda$-invariant for the cyclotomic $\mathbf Z_3$-extension of $F_0$. The theory of 3-adic $L$-functions gives conditions for $\lambda\ge 2$ in terms of $\epsilon_0$ and the class numbers of $F_0$ and $K_0$. We construct units of $K_1$, the first level of the $\mathbf Z_3$-extension of $K_0$, that potentially occur as Kummer generators of unramified extensions of $F_1(\zeta_3)$ and which give an algebraic interpretation of the condition that $\lambda\ge 2$. We also discuss similar results on $\lambda\ge 2$ that arise from work of Gross-Koblitz.