A proof of Siegel’s weight formula. (English) Zbl 0743.11023

Eskin, Alex; Rudnick, Zeév; Sarnak, Peter

Int. Math. Res. Not. 1991, No. 5, 65-69 (1991).

Let \(A\) and \(B\) be symmetric, nondegenerate half integral matrices of size \(m\times m\) and \(n\times n\) \((n\leq m)\) respectively. Consider the matrix equation (1) \(^ tXAX=B\). The number of integral solutions \(X\) \((m\times n\) matrices) of (1) is denoted by \(N(A,B)\). The orthogonal group \(O(A)\) of \(A\) is denoted by \(G\). Let \(X_ 1,\ldots,X_ \nu\) be a complete set of \(G(\mathbb{Z})\)-inequivalent integral solutions to (1) and \(H_ j\) be the stabilizer of \(X_ j\). Here \(G(\mathbb{Z})\) is the group of \(\mathbb{Z}\)-points in \(G\). Then a special case of Siegel’s weight formula is \[ \frac{1}{\operatorname{vol}(G(\mathbb{Z})\backslash G(\mathbb{R}))} \sum^\nu_{j=1}\operatorname{vol}(H_j(\mathbb{Z})\backslash H_i(\mathbb{R}))=\prod_p\mu_p, \]
where \(\mu_p\) is the \(p\)-adic density of the solution of (1).
First the paper begins with the sketch of the proof of Siegel’s weight formula for indefinite quadrics \(V_k=\{X\mid F(X)=k\}=\{X\mid{^tX}AX=k\}\), where the indefinite quadratic form \(F\) comes from an appropriate \(A\) and \(k\in\mathbb{Z}-\{0\}\). The proof is a combination of the Hardy-Littlewood method with the asymptotic counting of the contributions of each \(G(\mathbb{Z})\)-orbit in \(V_k(\mathbb{Z})\) by nonabelian harmonic analysis on \(G(\mathbb{Z})\backslash G(\mathbb{R})\). However full details are completely lacking here and they will appear elsewhere.
Next the paper shows that the special weight formula for indefinite quadrics obtained thus far implies the Tamagawa number \(\tau(G)\) equals 2 for all orthogonal groups \(G\). After M. Kneser and A. Weil, the authors remark that this fact yields Siegel’s general weight formula. This completes the proof.

Reviewer: Michio Ozeki (Hirosaki)

MSC:

11E45	Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11E12	Quadratic forms over global rings and fields

Keywords:

mass formula; Siegel’s weight formula; quadrics; Tamagawa number; orthogonal group