Extra Capitulation and Central Extensions

✍ Scribed by D. Folk

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 227 KB
• Volume: 50
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1995.1017

✦ Synopsis

Let (L) be a cyclic unramified extension of the number field (K), with (G:=) (\operatorname{Gal}(L / K)), and (L^{(1)}) the Hilbert class field of (L). The central object of studying those ideals of (K) which become principal, i.e., capitulate, has been (H^{1}\left(G, E_{L}\right)), where (E_{L}) denotes the group of global units of (L). However, if one lets (C_{L}) and (U_{L}) denote the idele class group of (L) and the group of unit ideles, respectively, there is an isomorphism (H^{i+1}\left(G, E_{L}\right)=H^{i}\left(G, U_{L} / E_{L}\right)), and (U_{L} / E_{L}) has the advantage of being isomorphic to an idele class subgroup of (C_{L}); this is our basic tool. In this paper, we study "extra" capitulation, that is, whenever there is more capitulation than one would normally expect. More precisely, we show that there is a nontrivial ramified central extension of (L^{11} M / K), with (M) some abelian extension of (K), exactly when there is extra capitulation. 1995 Academic Press. Inc.