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Consider an irreducible $\mathrm{mod}$ $p$ representation:

$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$

If $\rho$ is odd, it was conjectured by Serre in the 70s and proved by Khare-Wintenberger in the 00s that $\rho$ comes from a modular form.

But much like in the characteristic 0 case, for $\rho$ even the situation is even more complicated. In particular, it can be easily proved that they never arise form modular forms.

The question is, where are (irreducible) even mod p Galois representations conjectured to come from? What is the analogue of Serre's conjecture in the even case?

This issue has been mentioned several times on MO. For example, this 2012 question:

Is there a theory of Maaß forms over finite fields?, by Chandan Singh Dalawat.

asks something a bit weaker, whether those representations come from Maass forms, for which I'm quite sure the answer is no, but I wouldn't know how to justify it.

Kevin Buzzard briefly addressed this issue in his reply to this 2010 question:

Is there a canonical notion of “mod-l automorphic representation”?, by David Hansen.

And Jim Stankewicz in his answer to this other question:

The significance of modularity for all Galois representations, by Jonah Sinick

although it seems to be some confusion with mod l representations, see the comments.

The closest to an answer I've come across is this blog post, which seems to suggest that even representations are either dihedral or automorphic from $U(3)$, but I don't quite follow the argument.

Any information or reference would be appreciated.

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    $\begingroup$ Even mod $p$ Galois reps can come from modular forms if $p=2$! This is a terrifying thing because it means that given a non-holomorphic algebraic Maass form there should be a holomorphic one congruent to it mod 2. $\endgroup$
    – eric
    Jan 6, 2016 at 21:33
  • $\begingroup$ @eric That's weird and interesting! Do you know of any reference where it is discussed? $\endgroup$
    – Myshkin
    Jan 10, 2016 at 14:36
  • $\begingroup$ Serre remarks on it in his 1987 Duke paper where he formulates his conjecture, but only to the extent that he is surprised by the phenomenon and that Mestre had checked some numerical examples for him and verified they really were modular. $\endgroup$
    – eric
    Mar 16, 2016 at 23:27
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    $\begingroup$ This is not an answer, but Frank Calegari proved that any such representation must in fact have non-distinct Hodge-Tate weights, so that might help determine where these things are (or are not) coming from. math.uchicago.edu/~fcale/papers/EvenPotential.pdf $\endgroup$
    – Jeff H
    Jun 4, 2016 at 17:42

2 Answers 2

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To lift $\bar{\rho}$ to a geometric representation in the sense of Fontaine-Mazur, the standard technique requires that $\bar{\rho}$ is balanced, i.e. the dimension of a certain Selmer group must equal the dimension of its dual Selmer group (associated to a certain deformation problem). This is not the case for even representations. However, if one relaxes the $p$-adic Hodge theoretic condition at $p$ (crystalline or de Rham for example) then it becomes possible to sometimes lift an even $\bar{\rho}$ to a characteristic-zero representation $\rho:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{Z}_p)$ which is ramified at only finitely many primes as was demonstrated by Ramakrishna in "Deforming an Even Galois reprentation". In this paper, some even representations were lifted to characteristic zero for $p=3$. These lifts should not satisfy the de Rham condition at $p$. On the other hand, you're not expected to get geometric lifts even for small primes ($p>7$ was deduced in Calegari's paper).

If you're looking for lifts which satisfy a local condition at $p$ you will get lifts which are ramified at an infinite density zero set of primes. These may be constructed through an application of the lifting strategy of Khare, Larsen and Ramakrishna. You may be interested in knowing where such representations in general come from.

Also, it is worth noting that the standard geometric lifting technique does not apply for residual representations $\bar{\rho}:G_{K}\rightarrow \text{GL}_2(\mathbb{F}_{p^m})$ if $K$ is not totally real since in this case the associated Selmer and dual Selmer groups do not match up in dimension. This in no means implies that there are no geometric lifts when $K$ is imaginary quadratic (for instance).

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Even reps. typically won't admit geometric lifts from char. $p$ to char. $0$.

The blog post you link explains how, by applying a symmetric square functoriality, one can move these even mod $p$ reps. into a situation in which they can be lifted to char. zero. The weight $k$ that appears in that answer will then be related to the mod $p$ behaviour of (the symm. square of) the original $\overline{\rho}$ --- because this is how weights are determined in the context of Serre's conjecture.

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