Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0747.11055
Kurihara, Masato
Some remarks on conjectures about cyclotomic fields and $K$-groups of $\bbfZ$. (English)
[J] Compos. Math. 81, No.2, 223-236 (1992). ISSN 0010-437X; ISSN 1570-5846/e

For an odd prime $p$ let $A$ denote the $p$-Sylow-subgroup of the ideal- class-group of the $p$-th cyclotomic field $\bbfQ(\mu\sb p)$. The group $A$ decomposes into eigenspaces $A\sp{[k]}$, $0\le k<p-1$, with respect to the powers $\omega\sp k$ of the Teichmüller character $\omega$. The classical conjectures of Kummer-Vandiver predict that $A\sp{[k]}=0$ for even $k$, which in turn implies via reflection that $A\sp{[j]}$ is cyclic of order $L(0,\omega\sp{-j})$ for odd $j$. The author shows, that these conjectures are consequences of conjectures about the structure of Quillen's $K$-groups of the ring of integers $\bbfZ$. For instance, the vanishing of the $p$-primary torsion in $K\sb{4n}(\bbfZ)$, $n\ne 0$, implies the triviality of the eigenspace $A\sp{[i]}$ for $i\equiv-2n\bmod p-1$. In particular, since it is well-known that $K\sb 4(\bbfZ)$ has only 2- and 3-torsion, the eigenspace $A\sp{[p-3]}$ is trivial and $A\sp{[3]}$ is cyclic for any odd prime $p$. Some applications of these results are given, and furthermore the author shows in an Appendix that the concept of Kolyvagin's Euler systems works well for the study of the étale cohomology groups $H\sp 2\sb{et}(\bbfZ[1/p],\bbfZ\sb p(r))$ for odd numbers $r\ge 3$.
[M.Kolster (Hamilton/Ontario)]

MSC 2000:: *11R70 K-theory of global fields
11R18 Cyclotomic extensions
19F27 Etale cohomology, etc.

Keywords: Kummer-Vandiver conjecture; Quillen $K$-theory; Euler systems

Cited in: Zbl 1012.11094 Zbl 0855.11057 Zbl 0851.19003 Zbl 0797.11087 Zbl 0778.11066

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]