Did Smith correctly state the mass formula?

H.J.S. "normal form" Smith was the first, in 1867, to state the mass formula for integral quadratic forms in a genus of 4 or more variables. This was forgotten and the formula is usually attributed to Minkowski, who rediscovered it in 1885, and to Siegel, who corrected Minkowski in 1935. Conway and Sloane mention a number of erroneous sources after Siegel, but though they quote Smith worrying that he has made an error, they mention no errror, suggesting that he got it right. On the other hand, they say that in addition to the 1867 paper, Smith had an 1884 paper which won a prize from the French Academy jointly with Minkowski. So I expect that the judges compared the two results and would have noticed a discrepancy, suggesting that they were equally right or wrong.

I'm not sure what any of these papers actually state. The 1884 competition was about the specific case of the sum of five squares, so perhaps the entries imposed restrictions that saved their correctness and Minkowski's error was only in the 1885 extension? In particular, I think that he restricted to odd forms in 1884, but I forget my source for this. Conway and Sloane say that Smith's 1867 formula restricted to odd determinant.

More generally, what are good sources for the history of quadratic forms?


1 Answer 1


Not entirely sure about Smith. As far as i know, Conway and Sloane's version is correct, I've used it, but they give no proof at all. This is one reason that Shimura got involved. He and his student Jonathan Hanke both published on proofs of the mass formula. Shimura wrote a book, I think that must be item 22 at http://www.ams.org/journals/bull/2006-43-03/S0273-0979-06-01107-4/

Let's see. All the difficulty lies in the 2-adic contribution. There have been attempts to make a canonical 2-adic representative for quadratic forms. See J. W. S. Cassels, Rational Quadratic Forms. On page 120 we read

We do not attempt to specify a unique canonical form [see Jones(1944), Pall (1945), or Watson (1976a)]: that is more a job for a parliamentary draftsman than for a mathematician.

Note that C+S use Watson's version.

There is a fair amount of stuff at http://zakuski.math.utsa.edu/~kap/forms.html and http://zakuski.math.utsa.edu/~kap/more_than_this.html which tends to the modern.

Probably enough for now.


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