Let $\mathbb{ K}$ be an imaginary quadratic field such that $2$ splits into two primes $\mathfrak{ p}$ and $\bar{ \mathfrak{ p}}$. Let $\mathbb{ K}_{ \infty}$ be the unique $\mathbb{ Z}_2$-extension of $\mathbb{ K}$ unramified outside $\mathfrak{ p}$. Let $\mathfrak{ f}$ be an ideal coprime to $\mathfrak{ p}$ and $\mathbb{ L}$ be an arbitrary extension of $\mathbb{ K}$ contained in the ray class field $\mathbb{ K}(\mathfrak{ p}^2\mathfrak{ f})$. Let $\mathbb{ L}_{ \infty}=\mathbb{ K}_{ \infty}\mathbb{ L}$ and let $\mathbb{ M}$ be the maximal $p$-abelian, $\mathfrak{ p}$-ramified extension of $\mathbb{ L}_{ \infty}$. We set $X=Gal(\mathbb{ M}/\mathbb{ L}_{ \infty})$. In this paper we prove the Iwasawa main conjecture for the module $X$.
Categories: math.NT