9
$\begingroup$

I apologize since this is a quite vague question. And I am personally at an expert in these fields at all.

It seems to me that there are two main directions of the Langlands program, namely, construction of Galois representation, modularity (automorphy) theorems. However, in all that have been achieved, there is always the assumption that the "base field" is either totally real or CM. For example, the most recent Scholze's work on construction of Galois representation (as well as work of many other authors), or the recent automorphy lifting theorems by various authors.

Indeed, it seems to me all these work use the tool of "unitary Shimura varieties" a lot, which, is only available with CM base fields.

What is the difficulty in going beyond CM field? And what are the possible ways to attack them?

Thank you!

Improve this question
$\endgroup$
2
  • $\begingroup$ At a very elementary level, there are special algebraicity properties for GL$_1$ over CM fields, due to a theorem of Artin and Weil. See the recent PhD thesis of Stefan Patrikis for an extension of these algebraicity considerations to more general reductive groups. $\endgroup$
    – user76758
    Commented Dec 25, 2013 at 21:07
  • $\begingroup$ I have been wondering the same thing, but apparently some papers (for instance, the ones of Edward Frenkel, for example this one arxiv.org/abs/hep-th/0512172 ) did not say anything about totally real, CM-type and, furthermore do not require the representation to be geometric. I don't know if has been any advances on the theory or bypass these details or if it's just poor writing by the author. $\endgroup$
    – user40276
    Commented Jul 26, 2014 at 4:26

0

Reset to default

You must log in to answer this question.