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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.

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    $\begingroup$ Langlands-Tunnell aims at generalizing that if $L/\Bbb{Q}$ ($L=\Bbb{Q}(E[p])$) is Galois with dihedral Galois group and the quadratic subfield is imaginary then its Artin representations are modular from class field theory and the (weight $1$) modularity of Hecke L-series of $\Bbb{Q}(\sqrt{-d})$ (usually we start with $L=\Bbb{Q}(x),x^3\pm x+1=0$) $\endgroup$ – reuns Mar 8 '20 at 23:28
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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).

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  • $\begingroup$ Can there be solvable linear groups images of irreducible Galois representations on $E[p]$ for large primes? I have seen that Jordan and Mal'cev characterized such groups but can't deduce that they cannot receive such representations. I know little about elliptic curves. Are there cases of the Serre modularity conjecture which are "relatively easy" to prove for some elliptic representations, other than by Langlands--Tunnell, perhaps for those on Frey curves? Perhaps by methods unavailable in the 1990's. Thank you. $\endgroup$ – plm Mar 8 '20 at 23:27
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    $\begingroup$ If E is a non-CM elliptic curve, it is "very rare" for the image of the Galois rep $\bar\rho_{E, p}$ to be anything smaller than the full $GL_2(\mathbf{F}_p)$. It is known that given $E$ the image is this for all but finitely many $p$. It is conjectured that for any number field $K$ there is a constant $C$ such that if $p > C$ then $\bar\rho_{E, p}$ is surjective for every non-CM elliptic curve $E / K$. So the answer to your question is "conjecturally no". (There is no royal road to Serre's conjecture.) $\endgroup$ – David Loeffler Mar 9 '20 at 7:25
  • $\begingroup$ Is there a good reason to believe that the "Serre conjecture" at some (interesting?) prime is the hardest part of modularity conjectures for Galois representations on motives -I mean hardest knowing current lifting techniques? Does modularity of motives follow from Serre-type conjectures plus some reasonable improvement of known lifting results? Also, who made the big image conjecture you mention, is there a good review of what to expect for images of geometric Galois representations, perhaps with remarks on the differences between $\mathbb Z_p$ and $\mathbb F_p$ coefficients? Thank you. $\endgroup$ – plm Mar 10 '20 at 5:07
  • $\begingroup$ Earlier I wrote some nonsense as comment, which I deleted. I was confused about things, among which the Fontaine-Mazur conjecture. I may well have written more nonsense though... $\endgroup$ – plm Mar 10 '20 at 5:09
  • $\begingroup$ This is turning into one of those hydra-like MO threads where every answer generates two follow-up questions. MO is a good place for answering sharply defined questions. If you just want to gain some feel for the shape of a subject, then MO isn't well suited for that -- read some survey articles, or go talk to a human being. $\endgroup$ – David Loeffler Mar 10 '20 at 7:19

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