Let $$\displaystyle f(x_1,x_2,x_3) = a_1 x_1^2 + a_2 x_2^2 + a_3 x_3^2,$$ $$\displaystyle g(x_1, x_2, x_3) = b_1 x_1^2 + b_2 x_2^2 + b_3 x_3^2$$

be two integral ternary quadratic forms with $f$ positive definite. Let $p$ be an odd prime and suppose that $f$ and $g$ are not equivalent modulo $p$, i.e., there does not exist an integer $k$ such that

$$\displaystyle f(x_1, x_2, x_3) \equiv k g(x_1, x_2, x_3) \pmod{p}$$

for all $x_1, x_2, x_3 \in \{0, 1, \cdots, p-1\}$. We can further assume that $f,g$ are irreducible modulo $p$. Let $B$ be a positive number, and define

$$\displaystyle N_{f,g,p}(B) = \#\{(x,y,z) \in \mathbb{Z}^3 : f(x,y,z) \equiv g(x,y,z) \equiv 0 \pmod{p}, f(x,y,z) \leq B \}.$$

Put $N_f(B) = \#\{(x,y,z) \in \mathbb{Z}^3 : f(x,y,z) \leq B\}$. Is it true that for some $l \geq 2$ we have

$$\displaystyle \lim_{B \rightarrow \infty} \frac{N_{f,g,p}(B)}{N_f(B)} = \frac{1}{p^l}?$$

If so, what error term can be proved for the expression

$$\displaystyle N_{f,g,p}(B) = \frac{1}{p^l} N_f(B) + o(N_f(B))?$$