A note on Greenberg's conjecture for real abelian number fields
For an odd prime \(p\), let \(k\) be a real abelian field with \([k: \mathbb{Q}]\) prime to \(p\). According to Greenberg's conjecture, the Iwasawa invariant \(\lambda_p\) of the cyclotomic \(\mathbb{Z}_p\)-extension \(k_\infty\) of \(k\) should vanish. The authors present a criterion for the vanishing of the ``\(\chi\)-part'' of \(\lambda_p\), where \(\chi\) is a character of \(k\), under the assumption that the Iwasawa power series for the \(p\)-adic \(L\)-function \(L_p (s, \chi)\) is irreducible. In the case of a real quadratic field \(k= \mathbb{Q} (\sqrt {m})\) this leads to quite smooth criteria for \(\lambda_p =0\); for example, if \(p\) is either ramified or inert in \(k\), then \(\lambda_p\) vanishes if and only if the exponent of \(p\) in the \(p\)-adic regulator of \(k_n\) (the \(n\)th layer of \(k_\infty/ k\)) exceeds, for some \(n\geq 0\), an explicitly given bound depending only upon \(n\) and \(p\). As an application the authors give examples of \(\mathbb{Q} (\sqrt {m})\) with \(\lambda_3= 0\) for \(1< m< 10^4\), \(m\) squarefree and \(m\not\equiv 1\pmod 3\). The case in which 3 splits in \(\mathbb{Q} (\sqrt {m})\) has been studied by the second author and several other people; see, e.g., his joint article with \textit{T. Fukuda} [in Acta Arith. 69, No. 3, 277-292 (1995; Zbl 0828.11060)].
scientific article; zbMATH DE number 871770