Let \(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_N(\mathbf{C})\) be a continuous irreducible representation. Artin conjectured that the L-function \(L(\rho,s)\) is analytically continues to an entire function on \(\mathbf{C}\) (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 \(\rho\) is automorphic which implies, inter alia, that \( L(\rho,s) = L(\pi,s)\) for a cuspidal automorphic representation \(\pi\) for \(\mathrm{GL}(N)(\mathbf{Q})\).
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 \(L(\rho,s)\) 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 \(K^{\mathrm{gal}}/\mathbf{Q}\) be a Galois extension with Galois group \(S_5\), and suppose that complex conjugation in this group is equal to \((12)(34)\). Now suppose that \(\rho\) is a representation of \(\mathrm{Gal}(K^{\mathrm{gal}}/\mathbf{Q})\). We already know that \(L(s,\rho)\) is meromorphic, as proved by Brauer and Artin. One thing that can be proven is that, in the particular case above, \(L(s,\rho)\) is holomorphic in a strip \(\mathrm{Re}(s) > 1 – c\) for some constant \(c > 0\) 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 \(L(s,\rho)\) has a pole in the strip \(\mathrm{Re}(s) > 1 – c\), 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 \(\rho\) 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 \(\zeta_H(s)\) for a certain subfield \(H\) of \(K^{\mathrm{gal}}\) of degree twelve. (The definition of \(H\) was exactly what was swallowed up by Springer, so it’s not actually defined in the paper. To define it, note that \(S_5\) has a faithful representation on six points. There is a degree six extension \(E\) which is the fixed field of the stabilizer of a point; then \(H\) is the compositum of \(E\) and the quadratic extension inside \(K^{\mathrm{gal}}.\)) However, the actual argument produces a zero in an Artin L-factor of \(\zeta_H(s)\) which is not divisible by the Dirichlet L-function for the quadratic character of \(S_5.\) 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 \(L(\rho,s)\) of interest, one deduces that they are analytic on the strip \(\mathrm{Re}(s) > 1 – c\) where one can take
\(\displaystyle{1 – c = 1 – \frac{1}{4 \log |\Delta_H|}}.\)
The result of Stark, BTW, is why one could effectively solve the class number at most \(X\) problem for totally complex CM fields which were not imaginary quadratic fields before Goldfeld–Gross–Zagier.
Frank,
Your title reminds me of a great Dan Aykroyd skit on an early SNL, where he played a late night talk radio host, and kept proposing more and more outrageous topics in a (futile) effort to get someone to call in. But I’ll bite. What “applications” do you have in mind for the Riemann Hypothesis, or the conjecture of Birch and Swinnerton-Dyer, or the Hodge conjecture?
Dick
The *GRH* undisputedly has many interesting consequences (the other Artin’s conjecture, effective Brauer-Siegel, and many more). Even the classical RH has consequences concerning the approximation of $latex \pi(x)$ by $latex \mathrm{Li}(x)$. BSD? It does what it says on the bottle: take an elliptic curve and you can determine whether it has infinitely many points or not. The Hodge conjecture? Throw in the standard conjectures and one gets a robust theory of motives. If you wanted to give famous conjectures with absolutely zero interesting consequences, then surely additive number theory supplies the ne plus ultra of such problems: the Goldbach and twin prime conjectures.
Of course I agree with your implicit thesis that the purpose of proving theorems is to advance understanding rather than merely as a means to proving … more theorems. (It can safely be said that we learnt quite a lot from the proof of Fermat.) But doesn’t it seem a little sad that one can’t deduce anything from Artin’s conjecture? Unlike (say) in the case of elliptic curves, the results of Cebotarev and Artin-Brauer allow one to extract all of the relevant juices out of $latex L(\rho,s)$ without knowing that it is holomorphic. In part, this is because the critical values for Artin motives are on the edge of the strip, and one can happily talk about the special value $latex L(\rho,1)$ without having to know modularity.
I guess one consequence of a particular case of Artin’s conjecture is … more cases of Artin’s conjecture! More precisely, if one knows that $latex \rho$ and $latex \varrho$ satisfy Artin’s conjecture in the strong Langlands sense (= automorphic), then $latex \rho \otimes \varrho$ satisfies Artin’s conjecture in the weak sense, by Rankin-Selberg.
Ask them to publish an erratum acknowledging fault.
Done!
Just to defend the honour of the people in your field slightly: even if they were to read the paper, it would be the arxiv version that they read, or the version on your web-page, not Springer’s version.
When I think of applications of modularity of elliptic curves, I not only think of all the theoretical applications (Gross–Zagier, Kato, Skinner–Urban, … ) but also of the fact that the tables of modular elliptic curves are complete tables of elliptic curves, ordered by conductor.
Similar, I think of modularity for odd two-dim’l Artin reps. of Gal(Q-bar/Q) as meaning that the table of weight one forms gives a complete list of certain number fields. Maybe this doesn’t count as an
application, but it means something. It doesn’t seem that much less effective as a GL_2 class field theory than classical CFT does for (say) totally real fields, when one has no idea how big/small the various ray class groups are.
(Actually, the version on my web page until a few days ago was the official version, and so included the error…)
Threading the needle slightly, I both agree and disagree with you here. You are talking about a a very special case when you select odd $latex 2$-dimensional Artin representations. In almost every other context, the automorphic forms corresponding to Artin representations can’t actually be computed at all in an exact way. Using class field theory, for example, you can really compute whether any particular field has an extension with any particular solvable Galois group $latex G$ unramified outside a given finite set of primes. And yet we still don’t know Artin for solvable extensions of $latex \mathbf{Q}$. So while the existence (conjectural or otherwise) of a bijection between finite representations of Galois groups of number fields and certain automorphic forms is a beautiful one, it’s a stretch to say that it is “computable.”
Regarding BSD, I’ve always felt that the most important portion is the finiteness of Sha, which implies that the standard algorithm used to compute the Mordell-Weil group actually terminates. From this point of view, the importance of the L-function can be questioned in the elliptic curve context as well. However, people far wiser than I seem to believe that trying to get at Sha forces the L-function on you, one way or another.
Dear MK,
I both completely agree and disagree with what you say. First, I think that the finiteness of Sha is a really fundamental problem. On the other hand, I think your resulting corollary (that the MW group of E can be computed algorithmically) is not particularly interesting at all! I mean, as a practical matter, one can compute E(Q) anyway, and knowing that Sha is finite doesn’t help. It seems a little like saying that the finiteness of the class group is a fundamental problem because it allows one to algorithmically perform explicit computations in number fields (rather than just saying that it is a fundamental structural result). I’ll also push back slightly on other other aspects of BSD — proving BSD modulo finiteness of Sha would be a brilliant result.
On a different matter, I also think I don’t necessarily agree with your experts. It seems possible that one could prove Sha is finite without automorphy — and I even have my own wise expert to back me up. To discuss a related result, I don’t think that the Leopolodt conjecture will necessarily be proved using automorphic methods either.
Well, it’s a curious thing, the L-function. Anyway, I hope you agree to write the article.