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On the Iwasawa invariants of certain real abelian fields. (English) Zbl 0886.11060

Let \(k\) be a real abelian field and let \(p\) be an odd prime that does not split in \(k\). Greenberg’s conjecture predicts that the Iwasawa invariant \(\lambda_p\) of \(k\) is zero. Assuming some further conditions for the pair \((k,p)\), the authors find a necessary and sufficient condition for this to be true. For many classes of abelian fields their criterion can be transformed into a form which involves only rational arithmetic. In particular, such a rational criterion is provided for certain real quadratic fields. For these fields there exist extensive previous calculations of \(\lambda_3\); the authors are able to include some new cases, and they also calculate values of \(\lambda_5\) and \(\lambda_7\).
To get an idea of the criterion, consider the \(n\)th layer (\(n\geq 0\)), say \(k_n\), of the cyclotomic \(\mathbb{Z}_p\)-extension of \(k\). Let \(E_n\) and \(\mathcal U_n\) denote the unit group of \(k_n\) and the group of principal units in the (unique) non-archimedean completion of \(k_n\), respectively. Denote by \(\mathcal E_n\) the closure of \(E_n \cap \mathcal U_n\) in \(\mathcal U_n\). As usual, regard \(\mathcal E_n\) and \(\mathcal U_n\) as modules over \(\mathbb{Z}_p[[T]]\) and consider their components \(\mathcal E_n(\chi)\) and \(\mathcal U_n(\chi)\) for the even characters \(\chi\) of \(k\). Let \(\lambda_\chi\) denote the \(\chi\)-component of \(\lambda_p\). The authors’ result is based on the fact that \(\lambda_\chi\) vanishes if and only if, for some \(n\), \(\mathcal U_n(\chi)\) properly contains \(\mathcal E_n(\chi)\). To have control of \(\mathcal U_n(\chi)\) they give an explicit generator for it; this comes from a cyclotomic unit \(c_n\) defined by means of the norm of \(1 - \zeta _{f_n}\), where \(f_n\) is the conductor of \(k_n\) (\(\zeta _m\) is a primitive \(m\)th root of 1). The criterion is formulated in terms of \(c_n\), and its rational variant depends on an auxiliary prime \(l \equiv 1 \pmod {f_n}\). This variant resembles Vandiver’s famous criterion for the nondivisibility by \(p\) of the class number of the \(p\)th real cyclotomic field.

MSC:

11R23 Iwasawa theory
11R20 Other abelian and metabelian extensions
11R11 Quadratic extensions
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