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I am reading David Geraghty's paper, 'Modularity lifting theorems for ordinary Galois representations'(https://link.springer.com/article/10.1007/s00208-018-1742-4) and I have a related question, which, despite some search in the literature, remains puzzling to me (I should say that I am not quite familiar with this field of research and my question may be too naive to the experts): Geraghty's paper is for unitary groups that are compact at infinity. Are there any modularity lifting theorems for non-compact unitary groups? It seems many results on modularity liftings are for compact groups (except perhaps GL(2), GSp(4)?) What are the difficulties for non-compact ones? I am especially interested in the case of split unitary groups GU(n,n). Thanks in advance!

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You might like to read the introduction of Harris' 2013 Crelle paper "The Taylor-Wiles method for coherent cohomology" (see link). Here is an excerpt:

In practice, all the higher-dimensional results, with the exception of [GT] and [Pi], have been based on topological cohomology of zero-dimensional Shimura varieties. This is because the Taylor–Wiles method does not work well in the presence of torsion, and there are no general methods for comparing torsion in the cohomology of locally symmetric spaces to automorphic forms.

(The references [GT] and [Pi] refer to works of Genestier–Tilouine and Pilloni, both focussing on $GSp(4)$.) Harris goes on to write (emphasis mine):

We obtain no new results about Galois representations, and in fact I believe that practically everything one wants to say about automorphy of Galois representations can be obtained from the zero-dimensional case, as in [CHT], using Langlands functoriality for classical groups (see [A] and, in special cases, [CHLN]). Our purpose is rather to prove [freeness results for coherent cohomology as a Hecke module].

This was written a decade ago, and in the meantime the technology has got much better. So torsion in the cohomology of symmetric spaces is not quite so mysterious now as it was in 2013, and this has been a crucial input in recent dramatic advances in modularity over non-totally-real fields (e.g. the 10-author paper by Allen et al) or for non-regular weights (e.g. Boxer–Calegari–Gee–Pilloni). However, it definitely still remains true that the 0-dimensional case is much easier than the positive-dimensional one.

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  • $\begingroup$ Thanks David for all these clear explanations! I will have a closer look at these references. $\endgroup$
    – Zhan
    Sep 25 at 15:34

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