Circular distributions and Euler systems. II. (English) Zbl 1023.11056

Seo, Soogil

Compos. Math. 137, No. 1, 91-98 (2003).

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.

Reviewer: Cornelius Greither (Neubiberg)

MSC:

11R18	Cyclotomic extensions
11R23	Iwasawa theory

Keywords:

cyclotomic units; Euler systems; class numbers