Let be a continuous irreducible representation. Artin conjectured that the L-function is analytically continues to an entire function on (except for the trivial representation where the is a simple pole at one) and satisfies a functional equation of a precise shape. Langlands later had the profound insight to link this conjecture to functoriality in the Langlands program, which would additionally imply that is automorphic which implies, inter alia, that for a cuspidal automorphic representation for .
This is a beautiful and fundamental conjecture. However, it does appear to be completely useless for any actual applications. The most natural application of Artin’s conjecture is to prove … the Cebotarev density theorem. This is why Cebotarev’s density theorem is so amazing! True, one can upgrade the error estimates if one knows Artin, but to do this one also has to know GRH. And if you know GRH, you are not too far away from Artin anyway, because then at worst has poles on the critical strip, and so you can (essentially) get close to optimal bounds for Cebotarev anyway.
I thought a little bit about applications of Artin’s conjecture when I wrote a paper about it, but I came up empty. Then recently, I had occasion to look at my paper again, and found to my chagrin that when Springer made the final edit they lopped off a sentence in the statement of one of the main theorems. I guess that’s why the good people at Springer get paid the big bucks. (My best ever copy editing job, by the way, was for a paper in an AMS journal.) In a different direction, I guess it also reflects the deep study of this paper by people in the field that nobody has asked me about it. However, I did notice a statement in the paper that could be improved upon, which I will mention now.
To set the context, let be a Galois extension with Galois group , and suppose that complex conjugation in this group is equal to . Now suppose that is a representation of . We already know that is meromorphic, as proved by Brauer and Artin. One thing that can be proven is that, in the particular case above, is holomorphic in a strip for some constant which I described as “ineffective.” But looking at it again, I realized that it is not ineffective at all, due to a result of Stark. What one actually shows is that if has a pole in the strip , then there must also be another L-function for the same field which has a zero on the real line in this interval. Note that, again from by Artin, it is trivially the case that a pole of one L-function must come from the zero of another L-function, since the product of all such L-functions is the Dedekind zeta function. So the content here is that the offending pole has to be on the real line. One consequence is that, in any particular case, one can rigorously check that the L-function in question has no such zeros, and hence (combined with other results in this paper) that is automorphic. With help from Andrew Booker, I was able to compute one such example (Jo Dwyer has since gone on to compute a number of other examples.) On the other hand, back to the general case, we do have effective results for zeros on the real line! The result in the paper is stated in terms of the existence of a zero of for a certain subfield of of degree twelve. (The definition of was exactly what was swallowed up by Springer, so it’s not actually defined in the paper. To define it, note that has a faithful representation on six points. There is a degree six extension which is the fixed field of the stabilizer of a point; then is the compositum of and the quadratic extension inside ) However, the actual argument produces a zero in an Artin L-factor of which is not divisible by the Dirichlet L-function for the quadratic character of Stark shows (Some Effective Cases of the Brauer-Siegel Theorem) that such an L-function does not have Siegel zeros, and also gives an explicit estimate for the largest zero on the real line. In particular, for the of interest, one deduces that they are analytic on the strip where one can take
The result of Stark, BTW, is why one could effectively solve the class number at most problem for totally complex CM fields which were not imaginary quadratic fields before Goldfeld–Gross–Zagier.