Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits analytic continuation to the whole complex plane.

It is known for $1$-dimensional and induced representations, plus a few other special cases.

What is the status towards a proof? References would be very much appreciated.

Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math 170, Berlin, New York: Springer-Verlag, pp. 18–61.

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    $\begingroup$ For 2011 see mathoverflow.net/questions/59595/… $\endgroup$ – მამუკა ჯიბლაძე Jul 19 '16 at 5:25
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    $\begingroup$ More recently, Pilloni and Stroh have completed the proof of the Artin conjecture for two dimensional odd representations of the Galois group of any totally real field, see here $\endgroup$ – ulrich Jul 19 '16 at 6:31
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    $\begingroup$ Is that Pilloni-Stroh paper published? I think that Sasaki also announced a proof (after Pilloni-Stroh) but the last time I asked around (which admittedly was a while ago) it seemed that neither paper had been published. $\endgroup$ – znt Jul 19 '16 at 18:52
  • $\begingroup$ By the way, can an approach based on the Riemann surface on which the L-function is defined be interesting? $\endgroup$ – Sylvain JULIEN Jul 21 '16 at 14:25

This is the status as far as I know. For dimension $\leq 2$ it is up to date. For higher dimensional representations I'm sure it is very incomplete, so feel free to edit or comment.

Dimension 1. Known by "Artin-Hecke".

Dimension 2. Only open case is even $A_5$ representations. References for the known cases are:

  • $C_n$ "Artin-Hecke"

  • $D_n$ "Artin-Hecke" (see here)

  • $A_4$ Langlands, "Base change for GL(2)" (1980)

  • $S_4$ Tunnell, "Artin's conjecture for representations of octahedral type" (1981)

  • $A_5$ (odd $\rho$ over $\mathbb{Q}$) Khare-Wintenberger, "Serre's modularity conjecture (I)" (2009)

  • $A_5$ (odd $\rho$ over totally real fields) Pilloni-Stroh, "Surconvergence, ramification et modularité" (2013)

Dimension 3. Mostly wide open.

  • ($\rho$ induced) Jacquet, Piatetski-Shapiro, Shalika, "Relèvement cubique non normal" (1981)

  • ($\rho$ twist of a symmetric square) Gelbart, Jacquet, "A relation between automorphic representations of GL(2) and GL(3)" (1978)

Dimension 4. Only open solvable cases are $E_{2^4}\cdot D_{10}$ and $E_{2^4}\cdot F_{20}$. There are known non-solvable cases, but in general it's wide open.

  • ($\rho$ solvable) $\mathrm{GO}_4$ Ramakrishnan, "Modularity of solvable Artin representations of GO(4)-type" (2001)

  • ($\rho$ solvable) $E_{2^4}\cdot C_5$ Martin, "A symplectic case of Artin's conjecture" (2003)

Dimension $\geq 5$ Again there are some known cases, but mostly wide open.

The consensus is that a solution of the complete Artin conjecture is only accesible from general functoriality results such as base change or induction that would imply the strong Artin conjecture, and therefore Artin's holomorphy conjecture.

This means that the only way we have to prove that an Artin L-function $L(\rho, s)$ is holomorphic is to prove that $\rho$ is modular.

An alternative, as pointed out by Julien in the comments, would be to solve Selberg's orthogonality conjecture. That is implies the Artin conjecture is proved in "Selberg's Conjectures and Artin L-functions" (1994).

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    $\begingroup$ Or to prove Selberg's orthonomality conjecture, which is established under the assumption on the Ramanujan conjecture on the automorphic side. $\endgroup$ – Sylvain JULIEN Aug 2 '16 at 11:08
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    $\begingroup$ For the $A_5$ cases in dim 2, you need to be careful about the base field (not specified to be $\mathbb Q$ by OP). $\endgroup$ – Kimball Aug 3 '16 at 7:45

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