A specific problem on Class field theory

Question

Let $K$ be a quadratic complex number field. Let $p$ be a prime greater than $5$ unramified in $K/\mathbb{Q}$. Let $M$ be the compositum of all finite $p$- extensions of $K$ which are unramified outside the set of primes of $K$ lying above $p$. Let $M^{ab}$ be the maximal abelian extension of $K$ contained in $M$. Let $\Gamma^{ab}$ be Gal$(M^{ab}/K)$. For a prime $\mathfrak{P}$ of $K$ lying above $p$ let $U_{\mathfrak{P}}$ denote the local units of $K_{\mathfrak{P}}$, which is the completion of $K$ at prime $\mathfrak{P}$. Let $U_{1,\mathfrak{P}}$ denote the units that are congruent to $1$ mod $\mathfrak{P}$. Let $U_1=\prod_{\mathfrak{P}|p}U_{1,\mathfrak{P}}$. Let $H$ be the $p-$ Hilbert class field of $K$.

I want to show that (using class field theory) we have the following exact sequence

$1\rightarrow U_1 \xrightarrow{\beta} \Gamma^{ab} \xrightarrow{\alpha} Gal(H/K) \rightarrow 1$

It will be helpful if someone gives me a proof or a reference for this fact. The map $\alpha$ is obvious and it is also surjective but I do not know how to construct the map $\beta$ using class field theory and show that $\beta$ is injective and the sequence is exact. Thank you for help.

Answer 1

Let $H_n$ be the $p$-ray class field of $K$ of conductor $p^n$; then $M^{abs}$ is the union of all the $H_n$. The Galois group of $H_n/K$ is isomorphic to the $p$-ray class grip of $K$ of conductor $p^n$, and so your problem is to compute this.

In general, the ray class group of conductor $\mathfrak m$ sits in a short exact sequence $$1 \to \text{ units mod } \mathfrak m \,\, / \text{ global units} \to \text{ ray class group } \to \text{ class group } \to 1.$$ You want now want to take the $p$-primary part of this. Since $p > 3$, the global units in $K$ (which have order $2$, $4$, or $6$) have trivial $p$-primary part, while the $p$-primary part of the units mod $p^n$ is precisely the $1$-units mod $p^n$. Thus we get the s.e.s. $$ 1 \to 1\text{-units mod } p^n \to p\text{-ray class group of cond.} p^n \to p\text{-class group} \to 1.$$ Passing to the inverse limit in $n$ gives the s.e.s you asked about.

Answer 2

For any number field K, CFT gives an exact sequence (with your notations; $E$ is the group of global units):

$$E \otimes \mathbf Z_p \rightarrow U^1 \rightarrow \Gamma^{ab} \rightarrow Gal(H/K) \rightarrow 1.$$

This is shown in an idelic way in Washington's "Introduction to cyclotomic fields", capter 13, §13-1. The image of the leftmost map is the closure of $E$, embedded in $U^1$ diagonally. Leopoldt's conjecture asserts the injectivity of this map. It is proved for abelian $K$. For a complex quadratic $K$, $E$ is $C_2, C_4$ or $C_6$, hence is killed by tensorization by $\mathbf Z_p$ when $p > 5$ .