## 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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### References:

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