# Siegel's theorem

I want to learn the proof of the following theorem by Siegel. The statement of the theorem is taken from "Symmetric bilinear forms" by Milnor and Husemoller (pp. 44). They say that the proof is due to C.L. Seigel and appears in Gesammelte Abhandlungen I, Springer 1966. The problem is that I do not know German. Also, given that I am a theoretical computer scientist, I would like a proof which is as simple as possible.

Let $X$ be a positive definite bilinear form space of rank $\geq 2$ over $Z$. For each integer $k$ let $r_X(k)$ denote the number of distinct elements $x \in X$ satisfying the equation $x . x = k$. If the genus of $X$ contains only one isomorphism class then for $f(x_1,x_2, \dots, x_k) = x_1^2+x_2^2+\dots+x_n^2$

$r_x(k) = \epsilon \prod_{p=2,3,\dots,\infty} Df_p^{-1}(k)$

for every integer $k \neq 0$, where the coefficient $\epsilon$ is defined to be either $\frac{1}{2}$ or $1$ according as $n=2$ or $n > 2$.

Any pointers to a book/online report where the proof appears will be very helpful.

-
Have you checked out math.tifr.res.in/~publ/ln/tifr07.pdf – franz lemmermeyer Aug 28 '12 at 9:45
See Cassels' book (Rational quadratic forms) in the appendix, and Kitaoka's book (Arithmetic of quadratic forms) in Section 6.8. – emiliocba Aug 28 '12 at 13:08
@Franz: This one does not have it ... talks about the indefinite case. – cdubey Aug 31 '12 at 15:09
@Emili I am more interested in the $n\rightarrow \infty$ case. Small $n$ is not really interesting for the problem I am trying to solve. Kitaoka's book has it. Sadly, it is in Chapter 6 and he seems to be using a lot of results from the previous chapters. Is their an easier self contained proof for the $n\rightarrow \infty$ case ? – cdubey Aug 31 '12 at 15:10
I'm not sure what means the case $n\to\infty$. However, I don't know more references. – emiliocba Sep 1 '12 at 21:25