# What is Shimura referring to by “an incorrect formula given by Minkowski… known to most experts.”

In the article 'André Weil As I Knew Him' in the April 1999 issue of Notices of the AMS, Shimura recounts how in 1996 André Weil (then 90 years old) didn't remember a mistake of Minkowski. Specifically, quoting from mid-paragraph:

"... In fact, to check that point, I asked him whether Minkowski was reliable. He said, 'I think so.' At that point I realized that his recollection was faulty, since Minkowski gave an incorrect formula, as Siegel pointed out, and that was known to most experts. ..."

My question is:

Where specifically in Siegel and Minkowski's published works can I find the aforementioned formula (of Minkowski) and observation (of Siegel)?

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I believe he is referring to the result in Minkowski's dissertation that gives a formula for the mass (in the number/lattice theory sense) over a genus, that is, the sum of reciprocals of the group orders of all inequivalent quadratic forms in a genus. (wikipedia)

Siegel found and corrected the error in Minkowski's formula and also made many more generalizations in this Annals paper. Namely, there is an incorrect power of 2 in the formula.

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I do not know where in the original sources, but the topic under discussion is the mass formula for integral quadratic forms. The accepted source with correct information is Conway and Sloane,

Low Dimensional Lattices. IV. The Mass Formula

Proceedings of the Royal Society of London, A 419, 259-286 (1988).

In the actual publication, the tables are sprinkled throughout, and I found it difficult to read the text. I have some sort of preprint around here where the tables are all at the end, easier to find what you want. But it still takes some real patience, and in fact some imagination, to use properly.

I can recommend the book by the same Conway and Sloane, called Sphere Packings, Lattices, and Groups.

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Thanks very much for pointing out the accepted sources. –  Zavosh Mar 21 '10 at 5:15
This is also discussed (without proofs) in Milnor and Husemoller, "Symmetric bilinear forms". –  Victor Protsak Jun 30 '10 at 7:22