At the beginning of Milne's notes on class field theory, he has a quote by Emil Artin (as recalled by Mattuck in Recountings: Conversations with MIT mathematicians):

I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define L-series for non-abelian extensions. But for them to agree with the L-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I called it the General Reciprocity Law and tried to prove it but couldn't, even after many tries. Then I showed it to the other number theorists, but they all laughed at it, and I remember Hasse in particular telling me it couldn't possibly be true...

I assume he is referring to the definition of the Artin map as sending an unramified prime ideal $\mathfrak p$ to the Frobenius element $(\mathfrak p, L/K) \in \operatorname{Gal}(L/K)$ for an abelian extension $L/K$.

Questions: Why was it so hard for other number theorists to believe it? Artin probably talked to other people around 1924-27 (he says he took 3 years to prove the theorem in the same quote later) and Chebotarev had just recently proven his density theorem. Surely the density theorem provided strong evidence of the importance of the Frobenius element.

It is also something that one can explicitly verify at least in small cases and as Artin says, one can show that it also implies the other reciprocity laws proved by then. It seems very strange to outright dismiss the idea.

Were there other candidates for what the map should be or any reason to suspect that there should be no such canonical map? Surely there must have been strong reasons for such a strong rejection.

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    $\begingroup$ Interestingly, according to Shimura, Weil had doubts about (or saw little reason to believe in) the Shimura-Taniyama conjecture, which is a result of similar character. BTW, I think your question is most appropriate on this site, so don't worry about that. $\endgroup$
    – GH from MO
    Jul 3, 2016 at 18:45
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    $\begingroup$ This is pure speculation, but I think that Artin reciprocity is one of those things that seems "too good to be true." Any expert eventually develops, through long experience, an intuition as to what sorts of conjectures are too optimistic. For example, if I recall correctly, S.-T. Yau initially thought that mirror symmetry, as proposed by the physicists, seemed too good to be true. Of course, while the experts are usually right, they are sometimes wrong, and that's when we get some of the most electrifying discoveries in mathematics. $\endgroup$ Jul 3, 2016 at 20:17
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    $\begingroup$ As Lang (Notices Nov 1995) relates from Shimura, regarding the conversation of Weil's initial doubts: firstly, Taniyama opines that modular functions are themselves not enough, and then Weil says that he thinks the picture with elliptic curves in general should be completely different and mysterious (even more than uniformization by automorphic functions). Calling this the "Shimura-Taniyama conjecture" at this point in 1955 seems rather anachronistic. The later words at a IAS party (1962-4) by Weil are more direct and pointed, with the famous "both are denumerable" comment. $\endgroup$ Jul 4, 2016 at 1:30
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    $\begingroup$ Furthermore, Lang also indicates that Weil (1974) was annoyed by conjectures in general, particularly with the so-called Mordell conjecture due to there being not a shred of evidence for/against it (Lang queries the "so-called" here (as it being not due to Mordell?), but I think Weil just means that calling it a "conjecture" in the formal sense is not completely warranted w/o evidence). So I think that is more prevalent to his view, rather than anything about the modularity conjecture, or reciprocity. $\endgroup$ Jul 4, 2016 at 2:04

1 Answer 1


As GH has already remarked, the same thing happened a lot later after Taniyama and Shimura asked whether elliptic curves defined over the rationals are modular. To begin with your last question, there were no other candidates for the Artin isomorphism; reciprocity laws at the time were intimately connected to power residue symbols. Actually Euler had formulated the quadratic reciprocity law in the correct way (namely that the symbol $(\Delta/p)$ only depends on the residue class of $p$ modulo $\Delta$), but Legendre's formulation prevailed. I'm pretty sure that Hasse would not have laughed had Artin shown him right away that the general reciprocity law is equivalent to the known power reciprocity laws. Hasse did not have to wait for Chebotarev's density law to be convinced that Artin was right - by the time Chebotarev's ideas appeared it was clear that Artin must have been right. And, by the way, Chebotarev's article did a lot more than prove the importance of the Frobenius element - it provided Artin with the key idea for the proof of his reciprocity law, namely Hilbert's technique of abelian crossings.

Let me also add that in 1904, Felix Bernstein conjectured a reciprocity law in 1904 that is more or less equivalent to Artin's law in the special case of unramified abelian extensions. Its technical nature shows that it was not at all easy to guess a simple law such as Artin's from the known power reciprocity laws.

  • $\begingroup$ I didn't realize that the argument commonly known as "Chebotarev's field crossing argument" was in fact due to Hilbert! I checked your "Reciprocity Laws" book, but didn't see this discussed there. Could you say more about this? $\endgroup$ Jul 4, 2016 at 1:23
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    $\begingroup$ Hilbert used the basic idea in his proof of the Kronecker-Weber Theorem: for showing that an abelian extension $K/{\mathbb Q}$ is cyclotomic he looked at subfields of composita of $K$ with cyclotomic extensions. $\endgroup$ Jul 4, 2016 at 4:35
  • $\begingroup$ Thanks for clarifying that. I guess it still seems reasonable that the method is credited to Chebotarev, since it seems like a leap to go from crossing an abelian extension by a cyclotomic one to crossing a nonabelian by a cyclotomic. $\endgroup$ Jul 4, 2016 at 5:04
  • $\begingroup$ There are no nonabelian crossings in Chebotarev - the abelian case rules, as in Artin's reciprocity law. I don't think you can modify nonabelian extensions sufficiently using cyclotomic extensions, due to the lack of nontrivial subfields. $\endgroup$ Jul 4, 2016 at 17:53

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