Who first noticed that the Hilbert symbol is a Steinberg symbol ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:51:56Z http://mathoverflow.net/feeds/question/105791 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105791/who-first-noticed-that-the-hilbert-symbol-is-a-steinberg-symbol Who first noticed that the Hilbert symbol is a Steinberg symbol ? Chandan Singh Dalawat 2012-08-29T03:58:53Z 2012-09-19T23:45:59Z <p>Hilbert reformulated the quadratic reciprocity law of Gauß as a product formula $$\prod_v(a,b)_v=1$$ for the various local Hilbert symbols. For each place $v$ of $\bf Q$, the Hilbert symbol $(\ ,\ )_v$ is a <em>bimultiplicative</em> map $${\bf Q}_v^\times\times{\bf Q}_v^\times\to{\bf Z}^\times$$ so that, by definition, $(a,b)_v=1$ if and only if $a\in {\rm Im\;} N_b$ where $N_b$ denotes the norm map ${\bf Q}_v(\sqrt b)^\times\to{\bf Q}_v^\times$. An important property of the Hilbert symbol is that $$a+b=1\Longrightarrow (a,b)_v=1,$$ which makes it a Steinberg symbol. This property in not listed in older books such as Hasse's <em>Number theory</em> but it can be found in all modern treatments, such as Serre's <em>Course in arithmetic</em> or his <em>Local fields</em>, or Milnor's <em>K-theory</em>.</p> <p>I'm curious as to who first noticed that the Hilbert symbol is a Steinberg symbol. Was it Steinberg himself ? A precise reference will be appreciated.</p> http://mathoverflow.net/questions/105791/who-first-noticed-that-the-hilbert-symbol-is-a-steinberg-symbol/105802#105802 Answer by Srilakshmi for Who first noticed that the Hilbert symbol is a Steinberg symbol ? Srilakshmi 2012-08-29T07:52:31Z 2012-08-29T07:52:31Z <p>Matsumoto's computation of $K_2$ of a field in "Introduction to Algebraic K-Theory" by John Milnor. </p> http://mathoverflow.net/questions/105791/who-first-noticed-that-the-hilbert-symbol-is-a-steinberg-symbol/105818#105818 Answer by Jim Humphreys for Who first noticed that the Hilbert symbol is a Steinberg symbol ? Jim Humphreys 2012-08-29T11:52:02Z 2012-09-19T23:45:59Z <p>EDIT: After looking into the history more closely, I think it's fairly certain that the correct answer to the question is Calvin Moore. (See my added text below.) </p> <p>The question is interesting and looks straightforward, but it may not have a simple answer. A number of independent lines of research, differently motivated, converged miraculously in the late 1960s, so the word "noticed" in the question has to be placed in context. For example, one needs the notion of <em>topological</em> Steinberg symbol in the study of simple algebraic groups over various local fields. Here an essential contribution was made by Calvin Moore in his 1968 paper <em>Group extensions of p-adic and adelic linear groups</em> (IHES Publ. Math. 35).</p> <p>The history of all the related developments really ought to be told by one of the living participants. Seminars with Bass in that period got me interested for a while in the subject, leading me eventually to write an elementary introduction in the last part of my 1980 <em>Arithmetic Groups</em> (Springer Lect. Notes 789), where a lot of references are included; most of the key papers are now available online via numdam.org and such, while Steinberg's 1967-68 Yale lectures are posted on some webpages including his at UCLA and Bill Casselman's at UBC. But it would also help to know who was talking to whom during that crucial period.</p> <p>Briefly, Steinberg's work was purely algebraic at first and was motivated by Chevalley's 1955 Tohoku paper constructing versions of simple adjoint algebraic groups over arbitrary fields. In his 1962 Brussels conference paper, Steinberg worked out generators and relations in order to study non-adjoint groups and in particular see how projective modular representations of the adjoint groups would lift to "universal" groups. This led him to introduce what eventually became Steinberg symbols or cocycles. By the time of his Yale lectures, further connections were in the air. For instance, work on the Congruence Subgroup Problem (Bass, Lazard, Milnor, Serre) led Serre to an elegant formulation of the problem in terms of group extensions. Matsumoto's thesis provided more evidence of the close connection with Steinberg's formalism.</p> <p>Moore's work on the other hand came from his early interest in locally compact groups and their central extensions. Here he eventually found that a topological version of Steinberg's algebraic formalism would fit well with the classical ideas in number theory (local and global class field theory) explored in modern terms by Serre and others. There is a long and useful review of Moore's paper in Mathematical Reviews by Hyman Bass, if you have access.</p> <p>However the question in the header is answered, it should be kept in mind that the motivation for arriving at such a bizarre connection came from work of all these people. Connecting the dots required the existence of the dots.</p> <p>ADDED: A few more remarks about references, influences, timing. 1) Steinberg's 1962 Brussels paper (in French!) is reprinted along with his others in a single volume published by AMS and reasonably priced; but I don't know any accessible online source. However, most of his computations reappeared in his Yale lectures and in Matsumoto's thesis (with special cases treated in my lecture notes). 2) My best guess is that the combined work of Moore and Matsumoto filled in the connection with classical symbols and reciprocity laws. Note that Moore''s paper was submitted in January 1968 but lists Matsumoto's thesis in the references, while Matsumoto's thesis was submitted six months later and cites Moore's published paper. (The year of Moore's paper is either 1968 or 1969, depending on where you look.) Matsumoto thanks Bruhat, Serre, Samuel for their advice. On the other hand, Moore points to independent partial results by T. Kubota. He especially credits his conversations with a number of people including Bass and Serre. </p>