Converse to Modularity II: Maass cusp forms

Question

(This comes from this other question. You can find more details there)

The following bijection is now a theorem:

Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms

note: Galois representations are taken to be continuous, complex and linear.

It is natural to ask whether the equivalent correspondence for even representations holds:

Even irreducible 2-dim Galois repn $\longleftrightarrow$ Maass cusp forms with $\lambda = 1/4$

Every non-icosahedral even irr. 2-dim Galois representation is known to arise from a Maass cusp form with eigenvalue $1/4$, from the known cases of the strong Artin conjecture, so that

Even irreducible 2-dim Galois repn $\longrightarrow$ Maass cusp forms with $\lambda = 1/4$

is almost a theorem. You can assume that it is, for the purpose of this question: how far is this from being known to be a bijection, as in the odd case? Is there a converse theorem, analogous to the Serre-Deligne result?

Answer 1

This is a great question. I am no expert, but I share what I know.

We conjecture that this is a bijection. The most relevant publication that I am aware of is due to Blasius-Ramakrishnan (MR1012167) who provided a strategy to associate $2$-dimensional even complex Galois representations to Maass forms with Laplace eigenvalue $1/4$. Gelbart wrote in his AMS review to this paper that subsequently the authors together with Clozel and Harris proved the algebraicity of Hecke eigenvalues of such Maass forms unconditionally. Henniart gave a Bourbaki seminar on this topic (MR1040577), and then he published an erratum (MR1157812) whose AMS review by Zink reads:

The theorem of Blasius, Clozel and Ramakrishnan that the eigenvalues of Hecke operators for Maass forms of Galois type are algebraic numbers, which the author had discussed in the original paper, has to be considered unproved up to now. The problem is that the transfer $\Pi$ of a Maass representation $\pi$ from $\mathrm{GL}_2(\mathbf{A}_\mathbf{Q})$ to $\mathrm{GSp}_4(\mathbf{A}_\mathbf{Q})$ is unlike what has been stated before, namely its infinite component $\Pi_\infty$ occurs in an $L$-packet which does not contain limits of the discrete series.

The erratum is based on a letter by Blasius and Ramakrishnan, dated 9 April 1991.

Added. A similar question was asked at MO already, see here.