Greenberg's conjecture and the Iwasawa polynomial
The conjecture attributed to Greenberg, claiming that all lambda invariants \(\lambda_p\) of real fields \(k\) are zero, is difficult to attack, and there are not too many results available. A lot of work has been done, but extensive numerical results seem to have been obtained mainly for the invariant \(\lambda_3\) and \(k\) a real quadratic field of discriminant \(D\), say. In this context, let us mention \textit{J. Kraft} and \textit{R. Schoof} [Compos. Math. 97, 135-155 (1995; Zbl 0840.11043)] and \textit{T. Fukuda} and \textit{H. Taya} [Acta Arith. 69, 277-292 (1995; Zbl 0828.11060)]. These articles show that \(\lambda_3\) is zero for all but a few discriminants \(D<10000\). The author of the article under review previously developed a simple criterion which however only applies in restricted circumstances. The essential condition one needed in order to use this criterion is that the characteristic polynomial of the Iwasawa module \(Y\) is irreducible. Here \(Y\) is the Galois group of the maximal \(p\)-ramified pro-\(p\)-extension \(M\) of \(k\). In the paper under review, the author extends this criterion by dint of clever and involved algebraic computations to a more general setting. For details, we have to refer to the paper, in particular Theorem 3. The new criterion enables the author to settle some more cases of \(D<10000\). Meanwhile, the author and \textit{H. Ichimura} have found a new approach which allows to do \textit{all} cases \(D<10000\) [Tohoku Math. J., II. Ser. 49, 203-215 (1997; Zbl 0886.11060) and Int. J. Math. 7, 721-744 (1996; Zbl 0881.11075)].
scientific article; zbMATH DE number 1185524