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).

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# Siegel's Mass Formula for ternary indefinite quadratic forms

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 paper).

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).