Normal basis and Greenberg's conjecture
Let \(p\) be an odd prime number, \(F\) an imaginary quadratic field, \(k_ n\) the ray class field of \(F\bmod p^ n\) \((n\geq 1)\). Using modular units in the style of Taylor, Schertz and others [see e.g. \textit{M. J. Taylor}, Ann. Math., II. Ser. 121, 519-535 (1985; Zbl 0594.12008)], the author shows that the ring of \(p\)-integers of \(k_ n\) has a normal basis over that of \(k_ 1\). An easy consequence is that, for any \(\mathbb{Z}_ p\)- extension \(K/F\), the \(\mathbb{Z}_ p\)-extension \(k_ 1 K/k_ 1\) has a normal basis (meaning that the ring of \(p\)-integers of any finite extension in the tower has a normal basis over \(k_ 1\)). By a previous result of T. Fukuda and the author, if \(p=3\), \(F= \mathbb{Q} (\sqrt {-d})\), \(E= \mathbb{Q}( \sqrt{3d})\), \(({{-d} \over 3}) =-1\), \(3\mid h_ E\) and every \(\mathbb{Z}_ 3\)-extension of \(F\) has a normal basis, then the \(\lambda\)-invariant of \(E\) does not vanish. According to the author, this gives a slightly negative data against Greenberg's conjecture on the vanishing of the \(\lambda\)-invariant of any totally real number field. But because of the additional conditions, this reviewer does not think the data is relevant. Roughly speaking, the existence of normal bases as above means that a certain capitulation kernel in the cyclotomic \(\mathbb{Z}_ 3\)-extension of \(E\) vanishes [\textit{V. Fleckinger} and the reviewer, Manuscr. Math. 71, 183-195 (1991; Zbl 0732.11063)]. Then, modulo certain conditions on ramification (probably ensured by \(({{-d} \over 3})=-1 \)), the Iwasawa module upstairs would ``go down'' precisely on the 3-class group downstairs, and Greenberg's conjecture would be equivalent to \(3\nmid h_ E\). The situation is exactly the same as in the study of Vandiver's conjecture related to normal bases [\textit{I. Kersten} and \textit{J. Michaliček}, J. Number Theory 32, 371-386 (1989; Zbl 0709.11058)].
scientific article; zbMATH DE number 683138