Kummer generators and lambda invariants

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.