an:04150275
Zbl 0702.11022
Bayer, Pilar; Nart, Enric
Zeta functions and genus of quadratic forms
EN
Enseign. Math., II. S??r. 35, No. 3-4, 263-287 (1989).
00226925
1989
j
11E12 11E08 11E45
Gauss sums; integral quadratic forms; representation numbers; representation masses; genus
In his prize winning paper for the Paris Academy ``Grundlagen f??r die Theorie der quadratischen Formen mit ganzzahligen Koeffizienten'' (Ges. Abh. Bd. 1, 3-144) \textit{H. Minkowski} proved (in an appendix) that two nondegenerate integral quadratic forms of rank k, having the same discriminant and the same congruential representation numbers (and the same 2-type if \(k\geq 5)\) belong to the same genus. The present paper starts out with a reformulation of Minkowski's proof using modern terminology and removing the restriction on the discriminant. It is then shown how to translate this statement into the language of local (Section 2), adelic (Section 4) and global (Section 5) representation masses. As a consequence it is shown that the representation masses of the form f as well as those of the genus of f determine the genus of f with the same restricting conditions as above.
R.Schulze-Pillot