The Iwasawa \(\lambda\)-invariants of real abelian fields and the cyclotomic elements
Let \(p \neq l\) be primes and \(r\) an integer \(>1\). The author starts by defining integers \(n(r)_p\) in terms of elementary number theory, connects them with the \(p\)-class group of \(\mathbb{Q}(\zeta_p)\) and observes that Vandiver's conjecture would imply that \(n(r)_p = 1\) for all \(p\) and all odd \(r \geq 3\). He also observes that if the cyclotomic elements of Beilinson and of Deligne-Soulé coincide, then the supernatural number \(n(r) = \prod_p n(r)_p\) is actually an integer: \(n(r) < \infty\). Now let \(\omega\) be the Teichmüller character for \(\Delta_p = \text{ Gal}(\mathbb{Q}(\mu_p)/\mathbb{Q})\); for any \(\Delta_p\)-module \(M\), let \(M(\omega^i)\) be the submodule of \(M\) on which \(\Delta_p\) acts via \(\omega^i\). Let \(\mathbb{Q}_\infty = \mathbb{Q}(\mu_{p^\infty})\) denote the field generated by all \(p^n\)-th roots of unity (\(n \geq 1\)), let \(L_\infty\) be its maximal unramified prop-\(p\)-extension, and \(X = {\text Gal}(L_\infty/\mathbb{Q}_\infty)\); then \(\Lambda = \mathbb{Z}_p[[\text{ Gal}(\mathbb{Q}_\infty/\mathbb{Q})]](\omega^i)\) acts on \(X(\omega^i)\). The main theorem of the paper is the following: Fix a prime \(p\) and let \(r_0 \geq 3\) be an odd integer; then \(n(r)_p\) is bounded for \(r \equiv r_0 \bmod p-1\) if and only if the characteristic power series of \(X(\omega^{1-r_0})\) does not have a root in \(\mathbb{Z}_p\). The author also gives a similar theorem for the ``\(\chi\)-parts'' of \(n(r)\) for characters \(\chi\) such that \(\chi\omega^{1-r_0}(p) \neq 1\). In the last part of the paper, the results are applied to the verification of Greenberg's conjecture on the vanishing of Iwasawa's \(\lambda\)-invariant in some real quadratic number fields.
scientific article; zbMATH DE number 1423799