characteristic ideal of the Iwasawa module

Let $H$ be a complex biquadratic Galois extension of $\mathbb{Q}$ such that the galois group of $H$ is isomorphic to the Klein Group. Let $H_{\infty}$ be an anticyclotomic $\mathbb{Z}_p$-extension of $H$ and $L_{\infty}$ be the maximal abelian unramified $p$-extension of $H_{\infty}$. Assume that $p$ splits in $H_{+}$, where $H_{+}$ is the maximal totally real subfield of $H$ and $p$ doesn't totally split in $H$.

Let $X$ be the galois group of the extension $L_{\infty}/H_{\infty}$ and $X^{-}$ be the part of $X$ on which $\sigma \ne 1 \in \mathrm{Gal}(H_{+}/\mathbb{Q})$ acts by $-1$.

Can we relate the characteristic ideal of the Iwasawa module $X$ (resp. $X^{-}$) to some L $p$-adic functions ? Can we say anything about the $\mathbb{Z}_p$-rank of $X^{-}$ or $X$ ?

• I added the nt.number-theory tag (every question should have at least one arxiv subject class tag). I wouldn't have much hope for a positive answer to this question though -- as we discussed on one of your other recent questions, since $H / H_+$ doesn't satisfy Katz' ordinarity condition, there is no p-adic L-function available for $X$ or $X^-$ to be related to. Jul 12, 2016 at 8:42