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.


11R23 Iwasawa theory
11R20 Other abelian and metabelian extensions
11R11 Quadratic extensions
Full Text: DOI


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