On the Iwasawa lambda invariant of an imaginary abelian field of conductor \(3p^{n+1}\)
In the paper under review the authors computationally verify certain cases of Greenberg's conjecture on the vanishing of Iwasawa's \(\lambda\)-invariant. For a number field \(K\) let \(\lambda_K\) be the `standard' Iwasawa \(\lambda\)-invariant of the cyclotomic \(\mathbb Z_3\)-extension of \(K\). Let \(p \geq 5\) be a prime and consider the fields \[ K_n = \mathbb Q(\mathrm{cos}(2 \pi / p^{n+1}), \zeta_3) \] and the maximal totally real subfields \(K_n^+\). Then it is shown that for \(5 \leq p \leq 599\) one has \[ \lambda_{K_n} = \lambda_{K_0}, \quad n \geq 0 \] and \[ \lambda_{K_n^+} = 0, \quad n \geq 0. \] The latter is as predicted by Greenberg's conjecture. A similar result in the case \(p=2\) is also established.
scientific article; zbMATH DE number 6124195