In his paper "On the theory of indefnite quadratic forms", Siegel gives the formula (Thm. 1) $$ \mu(S,T)=\prod_p \alpha_p(S,T), $$ where
- $S$ is an $m\times m$ non singular integral symmetric matrix of signature $(r,m-r)$,
- $T$ is an $n\times n$ integral symmetric matrix,
- $\mu(S,T)$ is the measure of the representation of $T$ by $S$,
- $\alpha_p(S,T)$ is the $p$-adic density of the representation of $T$ by $S$ ($p$ over the rational primes),
and $$ n\leq r,\qquad n\leq m-r,\qquad 2(n+1) < m. $$
In my case, $n=1$ thus $T=t$ is just a (non zero) integer number, and $S$ has signature $(m-1,1)$ (or $(1,m-1)$). Hence, the theorem holds for $$ m>4. $$ However, in the fourth page, he added that the formula also holds for $m=4$.
I known that when $m=3$ ($S$ is a ternary quadratic form), the formula doesn't holds in general and strange things happen (see Borovoi's paperBOROVOI ).
My question is: Does the formula works for the following particular case?:
$$
S=
\begin{pmatrix}
A & \\
&-a
\end{pmatrix}
$$
where
- $A$ is a $2\times2$ integral positive definite symmetric matrix,
- $a$ is a positive integer, and
- $t$ is a negative integer.
I known that this is true for $A=I_2$ and $a=1$ (see this paper).
Thank you in advance.-.
