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First of all, let me apologize in advance for the terseness of this question.

It seems that by now there are well-developed techniques (the "Taylor-Wiles-Kisin" method) for proving modularity lifting theorems over totally real fields. For example, it is now known that many elliptic curves over totally real fields are modular. I am curious: what exactly is the stumbling block to proving such results over non-totally real fields? It seems to be common knowledge among experts that the usual techniques for comparing a universal deformation ring and a Hecke algebra in this situation break down quite badly, but I cannot find a reference for this in print. It would be great if someone could illustrate the problem here.

I understand that there are some obvious difficulties. For example, for $GL_2$ over an imaginary quadratic field, the locally symmetric spaces on which the relevant modular forms live is $SL_2(\mathbb{C})/SU(2)$, and arithmetic quotients of this are certainly not Shimura varieties. I am asking about more fundamental obstructions to applying the Taylor-Wiles-Kisin method, i.e strange behavior in the Hecke algebra and the universal deformation ring...

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# The difficulties in proving modularity lifting theorems over non-totally real fields

First of all, let me apologize in advance for the terseness of this question.

It seems that by now there are well-developed techniques (the "Taylor-Wiles-Kisin" method) for proving modularity lifting theorems over totally real fields. For example, it is now known that many elliptic curves over totally real fields are modular. I am curious: what exactly is the stumbling block to proving such results over non-totally real fields? It seems to be common knowledge among experts that the usual techniques for comparing a universal deformation ring and a Hecke algebra in this situation break down quite badly, but I cannot find a reference for this in print. It would be great if someone could illustrate the problem here.