nt.number theory - Is there an R=T type result for modular forms with additive reduction?

If you fix a prime $ell$, and consider the Galois action of the decomposition group $D_p$ on the (rational) $ell$-adic Tate module (here "rational" means tensored with $mathbb Q_{ell}$),
then (in a standard way, due to Deligne) you can convert this action into a representation
of the Weil--Deligne group, and so in particular of the Weil group. Restricting to the
inertia group, you get a representation of the inertia group $I_p$, known as the inertial type $tau$. It is independent of $ell$. (The only reason to detour through the Weil--Deligne group is to deal with possibly infinite image of tame inertia; if the elliptic curve has potentially good reduction, then we can skip this step and just take the representation of $I_p$ on the $ell$-adic Tate module, which has finite image and is independent of $ell$.)

[Added: In the above, one should insist that $ell neq p$. If $ell = p$, then one can also arrive at a Weil-Deligne representation, and hence
inertial type, which is the same as the one obtained as above for $ell neq p$, but to do this one must use Fontaine's theory: one forms the $D_{pst}$ of the rational $p$-adic Tate module,
which then can be converted into a Weil--Deligne representation in a standard way,
and hence gives an inertial type.]

Now one can look at the deformation ring $R_{rho}^{[0,1],tau}$ parameterizing lifts
of $rho$ of which at $p$ are of inertial type $tau$ and Hodge--Tate weights $0$ and $1$. (See Kisin's recent
JAMS paper about potentially semi-stable deformation rings.)

[Added: Here $ell = p$,
i.e. we are looking at $p$-adic deformations of $rho$ which are potentially semi-stable
at $p$, and whose inertial type, computed via $D_{pst}$ as in the above added remark,
is equal to $tau$. But note that, by the preceding discussion, these deformations do precisely capture the idea of lifts of $rho$ having the same "reduction type" as
the original elliptic curve $E$.]

Let's suppose that $E$ really does have potentially good reduction. Then Kisin's "moduli of finite flat group schemes" paper shows that any lift parameterized by $R^{[0,1],tau}$ is modular. This shows that $R^{[0,1],tau} = {mathbb T},$ for an appropriately chosen
${mathbb T}$.

One thing to note: unlike in the original Taylor--Wiles setting, from this statement one doesn't get quantitive information about adjoint Selmer groups, and one doesn't get any simple interpretation of what this $R = {mathbb T}$ theorem means on the integral level.
(In other words, Artinian-valued points of $R^{[0,1],tau}$ have no simple interpretation in terms of a ramification condition at $p$; this is related to the fact that the theory
of $D_{pst}$ only applies rationally, i.e. to ${mathbb Q}_p$-representations, not integrally,
i.e. not to representations over $mathbb Z_p$ or over Artin rings.)

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