I have been trying to understand a bit of the basics of deformations of Galois representations. One point which leaves me curious now is that proving modularity lifting with arbitrary ramification on the Tate module, compared to modulo $p$, seems quite alot to ask. As that is true, it is not so much after all. But is it not possible to find a prime $p$ for each, say semistable, elliptic curve $E$ where the representation on its Tate module $\rho\_{E,p}:G\_\mathbb Q\rightarrow End\_{\mathbb Z\_p}(T\_p(E))\cong GL(2,\mathbb Z\_p)$ has minimal ramification? That is, find a prime $p$ which does not divide the order of the minimal discriminant of $E$ at all $l|N\_E$, i.e. all primes of bad reduction of $E$ by the Néron-Ogg-Shafarevich criterion. Any large enough prime satisfies this condition so then my first question is: can you prove easily that the reduction modulo some large enough prime (or even an infinity of primes) $p$ is modular, as can be done for $p=3$ thanks to the Langlands-Tunnell theorem? This would then imply that $\rho\_{E,p}$, thus $E$, is modular without proving "$R\_\Sigma=T\_\Sigma$" for all $\Sigma$, thus "$R\_\text{unrestricted}=T\_\text{unrestricted}$".


Of course Wiles uses the primes 3 and 5 only for $\rho\_p$ so this prevents choosing the prime at will, and one has to assume ramification can increase arbitrarily on the Tate module. But did he or someone else after consider trying to use minimal modularity lifting only at some point?


If this is not possible for all semistable elliptic curves, for which elliptic curves would we expect it to be true? Could it be for Frey curves?


Are there Galois representations for which the induction argument used by Wiles to pass to arbitrary ramification on the liftings fails?


I hope you'll forgive me if this is easily found in standard references, or worse if I made a serious mistake, I find the subject quite demanding. Thanks.


EDIT: My first question didn't make much sense as it was, which proves I am a real newbie or the subject is really disturbing -probably both.



Modularity of elliptic curves with only minimal lifting
All the modularity-lifting theorems in the world aren't going to help you if you don't have something modular to lift! 


The reason $p=3$ is so important is that $GL\_2(\mathbf{F}\_3)$ is solvable, which allows you to use Langlands--Tunnell to show that the mod 3 representation is modular (at least if the image is large enough). Then Wiles' modularity lifting results allow you to get from this to full modularity. In order to use any other prime instead of 3, you'd need a variant of Langlands--Tunnell that worked for non-solvable Artin representations; and that's a massively harder problem than handling non-minimal ramification in modularity lifting (one which remains unsolved to this day).


