Strong Artin Conjecture

The strong Artin conjecture concerns an irreducible representation of the Galois group of a number field , given by a continuous complex Galois representation with finite image

The conjecture states that there is a cuspidal automorphic representation of whose unramified local parameters agree with the semisimple conjugacy class of at almost all places of . For a nontrivial , the conjecture implies that the Artin L-function is an entire function, as asserted by Artin's conjecture. The trivial 1-dimensional group representation is excluded from this holomorphy assertion. For , it gives the Riemann zeta function, which has a pole at .

Wang (2026) proved the strong Artin conjecture for 3-dimensional generalized octahedral representations, whose projective image is . This completes all 3-dimensional cases having solvable image. Wang (2026) also proved the strong Artin conjecture for every primitive solvable Artin representation of dimension 6.