Extra Capitulation and Central Extensions

Abstract

Let L be a cyclic unramified extension of the number field K, with G ≔ 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 H1(G, EL), where EL denotes the group of global units of L. However, if one lets CL and UL denote the idele class group of L and the group of unit ideles, respectively, there is an isomorphism Hi+1(G, EL) = Hi(G, UL/EL), and UL/EL has the advantage of being isomorphic to an idele class subgroup of CL; 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(1)M/K, with M some abelian extension of K, exactly when there is extra capitulation.