Greenberg's conjecture and relative unit groups for real quadratic fields
Let \(p\) be an odd prime and let \(k\) be a real quadratic field. Let \(\mathbb{Q}\subset \mathbb{Q}_1\subset \dots\subset \mathbb{Q}_{\infty}\) and \(k\subset k_1\subset \dots\subset k_{\infty}\) be the \(\mathbb{Z}_p\)-extensions of \(\mathbb{Q}\) and \(k\). Let \(\sigma\) be a topological generator of Gal\((k_{\infty}/k)\). Let \(E(k_n)\) be the unit group of \(k_n\). The author defines the relative unit group of \(k_n\) to be \[ E_{n,R}=\{\varepsilon\in E(k_n) \mid N_{k_n/\mathbb{Q}_n}(\varepsilon)=\pm 1, N_{k_n/k}(\varepsilon)=\pm 1\} \] and defines \(E_{n,R,p^n}=\{\varepsilon\in E_{n,R} \mid \varepsilon^{1+\sigma}\in E_{n,R}^{p^n}\}\). Let \(V_n=E_{n,R,p^n}/E_{n,R}^{p^n}\). The author shows that \(V_n\) has order \(p^n\), and that \(V_n\) is cyclic if and only if there exists \(\varepsilon\in E_{n,R}\) such that \(\varepsilon\) and its conjugates generate a subgroup of \(E_{n,R}\) of index prime to \(p\). Assume now that \(p\) does not split in \(k\). Let \(A_n\) be the \(p\)-Sylow subgroup of the ideal class group of \(k_n\). Let \(i_{0,n}:A_0\to A_n\) be the natural map. A result of \textit{R. Greenberg} [Am. J. Math. 98, 263-284 (1996; Zbl 0334.12013)] says that the Iwasawa invariant \(\lambda_p(k)=0\) if and only if \(i_{0,n}\) is the zero map for some \(n\geq 1\). The present author shows that if \(i_{0,n}\) is injective then \(V_n\) is cyclic. Conversely, if \(V_n\) is cyclic for all \(n\geq 1\) then \(i_{0,n}\) is injective for all \(n\geq 1\). This result is used to give some examples where \(\lambda_3(k)=0\). Now assume that \(p\) splits in \(k\). Let \(D_n\) be the subgroup of \(A_n\) generated by powers of classes of the primes above \(p\). Then \(A_n=D_n\) for all \(n\geq 0\) if and only if \(V_n\) is cyclic for all \(n\geq 1\). If Greenberg's conjecture that \(\lambda_3(k)=0\) is true and \(V_n\) is cyclic for all \(n\geq 1\), then \(A_n=D_n\) for all \(n\geq 0\).
scientific article; zbMATH DE number 1039113