On the main conjecture of Iwasawa theory for certain non-cyclotomic Z p -extensions

Abstract

We establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_\\infty)$ defined for a non-cyclotomic $\\mathbb{Z}_p$-extension $H_\\infty/H$, where $H$ is the Hilbert class field of an imaginary quadratic field $K$, at infinitely many primes $p$ which split in $K$ including $p=2$. As a consequence, we have that if $X(H_\\infty)=0$, the relevant $L$-values are $\\mathfrak{p}$-adic units, where $\\mathfrak{p}$ is a prime of $K$ above $p$. In addition, this paper provides an important step to the study of a main conjecture related to the conjecture of Birch and Swinnerton-Dyer, and to showing that $X(H_\\infty)$ is a finitely generated $\\mathbb{Z}_p$-module, which allows us to prove the weak $\\mathfrak{p}$-adic Leopoldt conjecture for a class of non-abelian extensions.