A note on \(\ell\)-parts of ray class groups (Q1087915)

Let \(\ell\) be an odd prime number and let k be an algebraic number field of finite degree. For an integer \(i>0\), let \(\zeta _ i\) denote a primitive \(\ell ^ i\)-th root of unity and put \(k_ i=k(\zeta _ i)\). For an ideal \({\mathfrak a}\) of k, let I(\({\mathfrak a})\) (resp. P(\({\mathfrak a}))\) denote the group of ideals (resp. principal ideals) of k prime to \({\mathfrak a}\), and \(P_{{\mathfrak a}}\) the ray ideal group of k modulo \({\mathfrak a}\), i.e., \(P_{{\mathfrak a}}=\{(x)|\) \(x\in k\), \(x\equiv 1 mod {\mathfrak a}\}\). Moreover let \(P'_{{\mathfrak a}}\) denote the group of elements of P(\({\mathfrak a})\) whose order modulo \(P_{{\mathfrak a}}\) is prime to \(\ell.\) The purpose of this note is to prove the following theorem. Assume \(\zeta _ 1\not\in k\) and \(k_ 1\neq k_ 2\). Let \[ 1\to N\to M\to I/P\to 1 \] be an abelian extension of the ideal class group I/P of k by a finite abelian \(\ell\)- group N. Then there exist infinitely many ideals S of k which satisfy the following: there is an isomorphism \(\Phi : I(S)/P_ S'\to M\) such that \(\Phi\) induces an isomorphism \(\Phi : P(S)/P_ S'\to N\) and the diagram \[ \begin{matrix} 1&\rightarrow&P(S)/P'_S&\rightarrow&I(S)/P'_S&\rightarrow&I/P&\rightarrow&1\\ &&\downarrow&&\downarrow\Phi&&\big|\big|\\ 1&\rightarrow&N&\rightarrow&M&\rightarrow&I/P&\rightarrow&1.\end{matrix} \] commutes. The proof depends on Kummer theory and uses Chebotarev's density theorem.