heiBOOKS

This manuscript arose from a course which I gave during the winter term 2016/17 at the University of Heidelberg. The attraction for me consisted in showing in detail that the formula of Minkowski-Siegel in Siegel‘s paper from 1935, is equivalent to the statement that the Tamagawa number of the orthogonal group (for a positive definite quadratic form) equals 2. Everybody knows that, but nobody has presented the calculations in detail. Furthermore, one can use the formulas to consider representations of numbers by quadratic forms. The starting point for this is taken from “Quadratic forms” by M. Kneser. Furthermore, by proving the Minkowski inequalities in orthogonal groups, I describe Siegel domains and estimate their volume.