Timeline for Who first noticed that the Hilbert symbol is a Steinberg symbol ?
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12 events
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| Nov 19, 2014 at 3:17 | comment | added | Venkataramana | @Humphreys: See also Proposition(3.1) of Bass-Milnor-Seree where they prove that the norm residue symbol is a Mennicke Symbol. | |
| Nov 14, 2014 at 15:51 | comment | added | Venkataramana | @Humphreys: It is possible that Bass-Milnor-Serre consider only the global Hilbert symbol and Mennicke symbol, but not the local Hilbert symbol and (Steinberg) symbol | |
| Nov 14, 2014 at 15:42 | comment | added | Venkataramana | @Humphreys: They use it all the time: that Mennicke symbol is the Hilbert symbol is used in computing the congruence subgroup kernel (e.g see Theorem (3.6) of Bass-Milnor Serre paper). The symbol $ (,)$ in Theorem (3.6) is the Hilbert symbol. | |
| Nov 14, 2014 at 15:06 | comment | added | Jim Humphreys | @Venkataramana: I don't know any real evidence in this direction. Certainly the Bass-Milnor-Serre work leaned heavily on Mennicke symbols, but these are not quite enough for the general problem. (Can you point out where Bass-Milnor-Serre identify the Mennicke symbol for type A groups with the Hilbert symbol?) A number of people were converging on the solution developed in Matsumoto's thesis, but I think Moore and Steinberg made the key discoveries. And certainly Serre was an important catalyst throughout; he might have his own version of what went on in that brief era of rapid progress. | |
| Nov 14, 2014 at 5:03 | comment | added | Venkataramana | For the group SL_n this was called the Mennicke symbol in Bass-Milnor-Serre paper and they identify it with hilbert symbol, therefore, maybe Mennicke noticed it . | |
| Sep 30, 2012 at 13:35 | comment | added | Jim Humphreys | @Chandan: Yes, the history is indeed complicated. But outside number theory the work of Steinberg (and later Moore) started instead with group extension problems and cocycles. It took a while to connect these apparently unrelated lines of thinking, which Moore seems to have been the first to do. | |
| Sep 24, 2012 at 15:58 | comment | added | Chandan Singh Dalawat | Yesterday I happened to come across Hasse's Tagebuch which Franz Lemmermeyer and Peter Roquette are editing. Today I noticed the entry 5.1 (dated 3.10.1927) which says that if $k$ is a number field containing a primitive $m$-th root of $1$, and if $\alpha,\beta,\gamma\in k^\times$ satisfy $\alpha+\beta=\gamma$, then $$ (\alpha,\beta)_m=(\alpha,\gamma)_m(\gamma,\beta)_m(−1,\gamma)_m $$ at every place of $k$; of course Steinberg's relation is the particular case $\gamma=1$. So Hasse was well aware of this relation in the 20s, and I wonder why he doesn't list it in his Number Theory. | |
| Sep 19, 2012 at 23:45 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| Sep 13, 2012 at 0:36 | comment | added | Jim Humphreys | @Chandan: I can update the history a bit, based on some email inquiries I made to participants: Moore was almost certainly the first person to discover the connection between Hilbert symbols and Steinberg cocycles/symbols, no later than early 1966. He gave a colloquium soon afterwards at UCLA with a title like Variations on a theme of Steinberg (with Steinberg in the audience). But everyone else I've mentioned played an essential role in completing the picture, especially the Congruence Subgroup Problem which Moore didn't consider. Very complicated history to reconstruct now. | |
| Aug 29, 2012 at 13:41 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| Aug 29, 2012 at 12:43 | comment | added | Chandan Singh Dalawat | Thank you very much for your remarks. It is to Steinberg's Brussels paper that I don't have access, but some of it is summarised in his Yale notes. What puzzles me is that everybody in the sixties (Moore, Steinberg, Matsumoto, Serre) seems to treat the Steinberg property of the norm residue symbol as being well known, and I can't find a place where somebody says : Look, what a beautiful property of the norm residue have I discovered ! | |
| Aug 29, 2012 at 11:52 | history | answered | Jim Humphreys | CC BY-SA 3.0 |