Irregular Lifts, Part II

This is the global counterpart to the last post. I was going to write this post in a more general setting, but the annoyances of general reductive groups got the better of me.

Suppose we fix the following:

1. A number field F and a prime p > 2.
2. A conjugacy class of involutions (possibly trivial) c_v of $\mathfrak{gl}_n$ for all real places of F.

Then does every representation $\overline{\rho}: G_F \rightarrow \mathrm{GL}_n(\overline{\mathbf{F}}_p),$

with complex conjugation acting on the adjoint by c_v for each real place of F

1. Have a de Rham lift?
2. Have a de Rham lift of non-regular weight?

I have basically come to the conclusion that the answer to this question is, almost always, no. Namely, the only time the answer is yes is when F is totally real and all the complex conjugations are totally odd. (With one caveat that comes later.)

Most of the theoretical evidence — slim that it is — is in favour of this minimalist conjecture. Namely:

• When n = 2 and F = Q and c is non-scalar, there is a global obstruction to lifting to a weight one modular form, since the image of such forms is a finite subgroup of GL_2(C), and this can already be precluded from making the image of $\overline{\rho}$ contain a large Borel subgroup.
• When n = 2 and p is totally split in F, there are also even local obstructions to lifting to non-regular weight. (There may be local obstructions in other cases as well, although I’m not sure.)
• When F has a real place, the usual conjectures imply that, when c_v is not the “odd” involution, there are obstructions to lifting to regular weights. In the extreme case when c_v is trivial, all lifts should be of trivial weight, and then one can prevent this happening by local (or conjecturally global) reasons similar to those mentioned above.

One can also extend this conjecture to other settings, where one still might conjecture the answer is always no unless one is in a context (regular weight) where $l_0 = 0.$

One caveat is that the case of GL(1) doesn’t quite work out. In this case, oddness is automatic and regularity is automatic, but even when F is not totally real there still exist lifts. I think this is too degenerate to really be so persuasive.

The first real case of this conjecture is when F is an imaginary CM field, and then the claim is that there should be representations $\overline{\rho}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbf{F}}_p),$

with no de Rham lifts. To be honest, I don’t have anything intelligent to say about how to prove this, I merely wanted to put on the record that I think I used to believe that such lifts might always exist, and now I’m willing to go on the record and conjecture that they don’t always exist. And, as I tell my group theory class, half the battle to answering a question in mathematics is determining what you think the right answer should be!

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