Extensions of homomorphisms and the structure of ray class groups (Q1320160)

Let \(A\), \(B\), \(C\) be abelian groups and \(f: C\to B\) be a homomorphism. Then an extension \((E, \varphi)\) of \(f\) by \(A\) is a commutative diagram of abelian groups \[ \begin{tikzcd} &&&C \ar[dl,"\varphi"']\ar[d,"f"] &\\ 0\ar[r] & A\ar[r] & E\ar[r] & B\ar[r] & 0\quad . \end{tikzcd} \] An isomorphism is defined on the set of extensions and the isomorphism classes of extensions of \(f\) by \(A\) are denoted by \(\text{Ext} (f; A)\). Such extensions are shown to occur in a natural way in class field theory where \(C=J\) is the idele group of the number field \(K\) and \(B= C_d\) is the ray class group of \(K\) modulo \(d\) and \(f\) is the canonical map from \(J\) to \(C_d\). The main result shows that for the canonical map \(f: \Sigma\to C_d\) there exist extensions \(M\) and \(L\) of \(K(\zeta)\) such that there is a canonical linear injection \[ \omega: \text{Gal} (M/L)\to \text{Ext} (f:\langle \zeta\rangle). \] Here \(\Sigma\) is a group closely related to the ideles of \(K\) and \(\zeta\) is a primitive \(m\)-th root of unity. The mapping \(\omega\) is defined using Artin symbols.