Timeline for First formulation of the Dedekind and Hasse-Weil conjectures
Current License: CC BY-SA 3.0
7 events
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| Sep 19, 2016 at 5:35 | comment | added | GH from MO | @user1952009: The above response was written by Vesselin Dimitrov, but let me give you some pointers. It is clear that $f(s):=\zeta_K(s)/\zeta_k(s)$ is meromorphic, while Aramata and Brauer independently proved that some positive integral power of $f(s)$ is holomorphic (as a product of certain Hecke $L$-functions associated with certain subfields of $K$). So $f(s)$ is meromorphic without a pole, hence it is holomorphic. For more details see the papers of Aramata (Proceedings of the Imperial Academy of Japan 9 (1933), 31-34) and Brauer (American Journal of Mathematics 69 (1947), 243-250). | |
| Sep 19, 2016 at 3:27 | comment | added | reuns | Since it is easy for normal extensions, can you sketch the proof :) ? @GHfromMO | |
| Nov 13, 2015 at 18:01 | comment | added | Vesselin Dimitrov | @Myshkin: On page 34 of Bombieri's article The classical theory of zeta and $L$-functions in the Milan Journal of Mathematics, the conjecture is attributed to Landau. So it probably appears in Landau's Handbuch, much earlier than in Artin. | |
| Oct 18, 2015 at 15:37 | comment | added | Vesselin Dimitrov | I didn't know about this; thanks for the reference! Yes, probably Dedekind thought of the general case, but it could be that this never appeared in print before Artin. Have you looked into Dedekind's Gesammelte mathematische Werke? | |
| Oct 18, 2015 at 8:39 | comment | added | Myshkin | I'm not sure that's it. The first time he published the proof was the in the 1900, but the result seems to be from 1873. See for example here, where van der Waall mentions the ealier date, but references the 1900 paper. I would be surprised if Dedekind didn't made the conjecture first, at least in a modest generality in the 60-70s, but I guess it's not entirely impossible that it was Artin 50 years later. | |
| Oct 18, 2015 at 3:33 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Oct 18, 2015 at 2:21 | history | answered | Vesselin Dimitrov | CC BY-SA 3.0 |