## The class group of $$\mathbb{Z}_p$$-extensions over totally real number fields.(English)Zbl 0896.11044

Let $$k$$ be a totally real number field, $$p$$ an odd prime, and $$k_\infty/k$$ a $$\mathbb{Z}_p$$-extension. Let $$L_\infty/k_\infty$$ be the maximal unramified abelian pro-$$p$$-extension, and $$M_\infty/k_\infty$$ the maximal abelian pro-$$p$$-extension unramified outside $$p$$. The first theorem of this paper states that if $$p$$ splits completely in $$k$$ and is totally ramified in $$k_\infty/k$$, then the following conditions are equivalent: i) $$L_\infty \neq k_\infty$$; ii) the $$p$$-class number of the $$n$$th layer $$k_n$$ of $$k_\infty/k$$ is nontrivial for every $$n \geq 1$$; iii) $$\zeta_p(0)$$ is a $$p$$-adic integer, where $$\zeta_p(s)$$ is the $$p$$-adic zeta function of $$k$$. Now assume in addition that Leopoldt’s conjecture holds for $$k$$ and $$p$$, then the author’s second theorem shows the equivalence of the following three assertions: i) there is nontrivial capitulation in $$k_\infty/k_1$$; ii) there is nontrivial capitulation in $$k_\infty/k_n$$ for some $$n \geq 0$$; iii) $$M_\infty \neq L_\infty$$. If, in addition, the $$p$$-class groups of the $$k_n$$ are cyclic and if the $$\mathbb{Z}_p$$-rank of $$M_\infty/k_\infty$$ is at least $$2$$, then Greenberg’s conjecture for $$k_\infty/k$$ is true.

### MSC:

 11R23 Iwasawa theory 11R29 Class numbers, class groups, discriminants
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### References:

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