Note: This is a fairly precise and detailed question about an important but technical aspect of algebraic number theory. My answer is written at a level that I think is appropriate for the question; it assumes some familiarity with the topic at hand.

The most basic difficulty is that there is non a map $R \rightarrow {\mathbb T}$ in general (i.e. one typically doesn't know how to create Galois representations attached to automorphic forms).

The second difficulty is that in the TWK method, one must argue with auxiliary primes (the primes typically labelled $Q$), and show that as you add these primes, ${\mathbb T}$ grows in a reasonable way (basically, is free over $\mathcal O[\Delta_Q],$ where $\Delta_Q$ is something like the $p$-Sylow subgroup of $({\mathbb Z}/Q{\mathbb Z})^{\times}.)$

One shows this (or some variant of it) by considering the analogous queston about cohomology of the arithmetic quotients. Suppose for a moment we are in the Shimura variety context, or perhaps the compact at infinity context. Then it will be the middle dimensional cohomology that is of interest, and if we localize at a non-Eisenstein maximal ideal we might hope to kill all other cohomology. Then we can replace a comuptation of middle dimensional cohomology by an Euler characteristic computation, and its easy to see that the Euler char. will multiply by $|\Delta_Q|$ when we add the auxiliary primes $Q$.

But in more general contexts, there won't be a single middle dimension in which the maximal ideal of interest is supported (even if it is non-Eisenstein), and computing Euler characteristics will just give $0$, which is not much use. It's not clear that it's even true that adding the auxiliary primes forces the approriate growth of cohomology, and possible torsion in the cohomology just adds to the complication.

There is much current work, by various groups of researchers, with various different approaches, aimed at breaking this barrier.

I should add that one can now handle certain questions about non-totally real field, say question related to conjugate self-dual Galois reps. over CM fields, because these are still related to a Shimura variety context. This plays a role in the recent progress on Sato--Tate for higher weight forms by Barnet-Lamb--Geraghty--Harris--Taylor and Barnet-Lamb--Gee--Geraghty, and is also the basis for a recent striking theorem of Calegari showing that if $\rho:G_{\mathbb Q} \to GL_2({\mathbb Q}_p)$ is ordinary at $p$ and de Rham with distinct Hodge--Tate weights (and probably $\overline{\rho}$ should satisfy some technical conditions), then $\rho$ is necessarily odd!

Note: This is a fairly precise and detailed question about an important but technical aspect of algebraic number theory. My answer is written at a level that I think is appropriate for the question; it assumes some familiarity with the topic at hand.

The most basic difficulty is that there is not a map $R \rightarrow {\mathbb T}$ in general (i.e. one typically doesn't know how to create Galois representations attached to automorphic forms).

The second difficulty is that in the TWK method, one must argue with auxiliary primes (the primes typically labelled $Q$), and show that as you add these primes, ${\mathbb T}$ grows in a reasonable way (basically, is free over $\mathcal O[\Delta_Q],$ where $\Delta_Q$ is something like the $p$-Sylow subgroup of $({\mathbb Z}/Q{\mathbb Z})^{\times}.)$

One shows this (or some variant of it) by considering the analogous queston about cohomology of the arithmetic quotients. Suppose for a moment we are in the Shimura variety context, or perhaps the compact at infinity context. Then it will be the middle dimensional cohomology that is of interest, and if we localize at a non-Eisenstein maximal ideal we might hope to kill all other cohomology. Then we can replace a computation of middle dimensional cohomology by an Euler characteristic computation, and its easy to see that the Euler char. will multiply by $|\Delta_Q|$ when we add the auxiliary primes $Q$.

But in more general contexts, there won't be a single middle dimension in which the maximal ideal of interest is supported (even if it is non-Eisenstein), and computing Euler characteristics will just give $0$, which is not much use. It's not clear that it's even true that adding the auxiliary primes forces the approriate growth of cohomology, and possible torsion in the cohomology just adds to the complication.

There is much current work, by various groups of researchers, with various different approaches, aimed at breaking this barrier.

I should add that one can now handle certain questions about non-totally real field, say question related to conjugate self-dual Galois reps. over CM fields, because these are still related to a Shimura variety context. This plays a role in the recent progress on Sato--Tate for higher weight forms by Barnet-Lamb--Geraghty--Harris--Taylor and Barnet-Lamb--Gee--Geraghty, and is also the basis for a recent striking theorem of Calegari showing that if $\rho:G_{\mathbb Q} \to GL_2({\mathbb Q}_p)$ is ordinary at $p$ and de Rham with distinct Hodge--Tate weights (and probably $\overline{\rho}$ should satisfy some technical conditions), then $\rho$ is necessarily odd!

Note: This is a fairly precise and detailed question about an important but technical aspect of algebraic number theory. My answer is written at a level that I think is appropriate for the question; it assumes some familiarity with the topic at hand.

The most basic difficulty is that there is non a map $R \rightarrow {\mathbb T}$ in general (i.e. one typically doesn't know how to create Galois representations attached to automorphic forms).

The second difficulty is that in the TWK method, one must argue with auxiliary primes (the primes typically labelled $Q$), and show that as you add these primes, ${\mathbb T}$ grows in a reasonable way (basically, is free over $\mathcal O[\Delta_Q],$ where $\Delta_Q$ is something like the $p$-Sylow subgroup of $({\mathbb Z}/Q{\mathbb Z})^{\times}.$

One shows this (or some variant of it) by considering the analogous queston about cohomology of the arithmetic quotients. Suppose for a moment we are in the Shimura variety context, or perhaps the compact at infinity context. Then it will be the middle dimensional cohomology that is of interest, and if we localize at a non-Eisenstein maximal ideal we might hope to kill of all other cohomology. Then we replace a comuptation of middle dimensional cohomology by an Euler characteristic computation, and its easy to see that the Euler char. will multiply by $|\Delta_Q|$ when we add the auxiliary primes $Q$.

But in more general contexts, there won't be a single middle dimension in which the maximal ideal of interest is supported (even if it is non-Eisenstein), and computing Euler characteristics will just give $0$, which is not much use. It's not clear that its even true that adding the auxiliary primes forces the approriate growth of cohomology, and possible torsion in the cohomology just adds to the complication.

There is much current work, by various groups of researchers, with various different approaches, aimed at breaking this barrier.

I should add that one can now handle certain questions about non-totally real field, say question related to conjugate self-dual Galois reps. over CM fields, because these are still related to a Shimura variety context. This plays a role in the recent progress on Sato--Tate for higher weight forms by Barnet-Lamb--Geraghty--Harris--Taylor and Barnet-Lamb--Gee--Geraghty, and is also the basis for a recent striking theorem of Calegari showing that if $\rho:G_{\mathbb Q} \to GL_2({\mathbb Q}_p)$ is ordinary at $p$ and de Rham with distinct Hodge--Tate weights (and probably $\overline{\rho}$ should satisfy some technical conditions), then $\rho$ is necessarily odd!

Note: This is a fairly precise and detailed question about an important but technical aspect of algebraic number theory. My answer is written at a level that I think is appropriate for the question; it assumes some familiarity with the topic at hand.

The most basic difficulty is that there is non a map $R \rightarrow {\mathbb T}$ in general (i.e. one typically doesn't know how to create Galois representations attached to automorphic forms).

The second difficulty is that in the TWK method, one must argue with auxiliary primes (the primes typically labelled $Q$), and show that as you add these primes, ${\mathbb T}$ grows in a reasonable way (basically, is free over $\mathcal O[\Delta_Q],$ where $\Delta_Q$ is something like the $p$-Sylow subgroup of $({\mathbb Z}/Q{\mathbb Z})^{\times}.)$

One shows this (or some variant of it) by considering the analogous queston about cohomology of the arithmetic quotients. Suppose for a moment we are in the Shimura variety context, or perhaps the compact at infinity context. Then it will be the middle dimensional cohomology that is of interest, and if we localize at a non-Eisenstein maximal ideal we might hope to kill all other cohomology. Then we can replace a comuptation of middle dimensional cohomology by an Euler characteristic computation, and its easy to see that the Euler char. will multiply by $|\Delta_Q|$ when we add the auxiliary primes $Q$.

But in more general contexts, there won't be a single middle dimension in which the maximal ideal of interest is supported (even if it is non-Eisenstein), and computing Euler characteristics will just give $0$, which is not much use. It's not clear that it's even true that adding the auxiliary primes forces the approriate growth of cohomology, and possible torsion in the cohomology just adds to the complication.

There is much current work, by various groups of researchers, with various different approaches, aimed at breaking this barrier.

I should add that one can now handle certain questions about non-totally real field, say question related to conjugate self-dual Galois reps. over CM fields, because these are still related to a Shimura variety context. This plays a role in the recent progress on Sato--Tate for higher weight forms by Barnet-Lamb--Geraghty--Harris--Taylor and Barnet-Lamb--Gee--Geraghty, and is also the basis for a recent striking theorem of Calegari showing that if $\rho:G_{\mathbb Q} \to GL_2({\mathbb Q}_p)$ is ordinary at $p$ and de Rham with distinct Hodge--Tate weights (and probably $\overline{\rho}$ should satisfy some technical conditions), then $\rho$ is necessarily odd!