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(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?

(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?

(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, equivalent to Serre-Deligne result?

(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?