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Circular distributions and Euler systems. II. (English) Zbl 1023.11056
This short article presents complements and variations on the author’s results “Circular distributions and Euler systems” [J. Number Theory 88, 366-379 (2001; Zbl 0995.11060)]. The question is again how far the module $${\mathfrak E}$$ of all circular distributions deviates from the Galois span $$R\Phi$$ of the canonical cyclotomic distribution $$\Phi$$. The naive equality $${\mathfrak E}=R\Phi$$ is not true, but Coleman conjectures that $${\mathfrak F}=R\Phi$$ where $${\mathfrak F}$$ is a submodule of $${\mathfrak E}$$ defined by quite natural congruence conditions. The author gives two theorems:
(A) If $$p$$ is coprime to $$\phi(n)$$, then the indices $$[{\mathfrak E}(np^r):C(np^r)]$$ are all prime to $$p$$ and bounded for $$1\leq r\to \infty$$, where $$C(m)$$ denotes the group of cyclotomic units in $$\mathbb{Q}(\zeta_m)$$.
(B) If $$p$$ is an odd prime and if we restrict our distributions to $$p$$-power roots of unity, then every circular distribution can be made cyclotomic by raising it to a nontrivial natural power.
The methods of proof are fairly standard and use the Main Conjecture again.

##### MSC:
 11R18 Cyclotomic extensions 11R23 Iwasawa theory
##### Keywords:
cyclotomic units; Euler systems; class numbers
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