Greenberg’s conjecture for Dirichlet characters of order divisible by \(p\). (English) Zbl 1061.11062

For any even Dirichlet character \(\chi\) and any prime \(p\), it is conjectured that the Iwasawa \(\lambda\)-invariant \(\lambda_{p,\chi}\) of the \(\chi\)-part of the ideal class group is zero [R. Greenberg, Am. J. Math. 98, 263–284 (1976; Zbl 0334.12013)]. This is often called Greenberg’s conjecture.
Let \(p\) be an odd prime number. In [T. Tsuji, Trans. Am. Math. Soc. 355, No. 9, 3699–3714 (2003; Zbl 1038.11072)], the author gave sufficient conditions for \(\lambda_{p,\chi}\) to be zero.
In the paper under review, the author considers these conditions and shows under some assumptions that there exist infinitely many characters \(\chi\) of order divisible by \(p\) for which Greenberg’s conjecture is true (Propositions 5 and 6) by using Kida’s formula due to W. M. Sinnott [Compos. Math. 53, 3–17 (1984; Zbl 0545.12011)].


11R23 Iwasawa theory
11R20 Other abelian and metabelian extensions
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