  Let $p$ be an odd prime and $f$ be a nearly ordinary Hilbert modular Hecke
eigenform defined over a totally real field $F$. Let $\mathbb{I}$ be an
irreducible component of the universal nearly ordinary or locally cyclotomic
deformation of the representation of $\mathrm{Gal}_F$ that is associated to
$f$. We study the deformation rings over a $p$-adic Lie extension $F_\infty$
that contains the cyclotomic $\mathbb{Z}_p$-extension of $F$. More precisely,
we prove a control theorem about these rings. We introduce a category
$\mathfrak{M}_{\mathcal H}^{\mathbb I}(\mathcal G)$, where $\mathcal
G=\mathrm{Gal}(F_\infty/F)$ and $\mathcal H=\mathrm{Gal}(F_\infty/F_{cyc})$,
which is the category of modules which are torsion with respect to a certain
Ore set, which generalizes the Ore set introduced by Venjakob. For Selmer
groups which are in this category, we formulate a Main conjecture in the spirit
of Noncommutative Iwasawa theory. We then set up a strategy to prove the
conjecture by generalizing work of Burns, Kato, Kakde, and Ritter and Weiss.
This requires appropriate generalizations of results of Oliver and Taylor, and
Oliver on Logarithms of certain $K$-groups, which we have presented here.

Number Theory