Let $f(x,y) = ax^2 + bxy + cy^2$ be a binary quadratic form with integer coefficients and non-zero discriminant $\Delta(f)$. It is well-known that $f$ is $\operatorname{GL}_2(\mathbb{R})$-equivalent (via substitution) to either $x^2 + y^2$ or $xy$, depending on the sign of $\Delta(f)$. Is there an analogue for this statement when the local field $\mathbb{R}$ is replaced with $\mathbb{Q}_p$ for some finite prime $p$, or the $p$-adic integers $\mathbb{Z}_p$? That is, does there exist for each prime $p$ a positive integer $k_p$ such that every binary quadratic form $f$ with integer coefficients is $\operatorname{GL}_2(\mathbb{Z}_p)$-equivalent to one of $k_p$ binary quadratic forms with integer coefficients?

Let $f(x,y) = ax^2 + bxy + cy^2$ be a binary quadratic form with integer coefficients and non-zero discriminant $\Delta(f)$. It is well-known that $f$ is $\operatorname{GL}_2(\mathbb{R})$-equivalent (via substitution) to either $x^2 + y^2$ or $xy$, depending on the sign of $\Delta(f)$. Is there an analogue for this statement when the local field $\mathbb{R}$ is replaced with $\mathbb{Q}_p$ for some finite prime $p$(or the $p$-adic integers $\mathbb{Z}_p$)? That is, does there exist for each prime $p$ a positive integer $k_p$ such that every binary quadratic form $f$ with integer coefficients is $\operatorname{GL}_2(\mathbb{Q}_p)$-equivalent to one of $k_p$ binary quadratic forms with integer coefficients?

Let $f(x,y) = ax^2 + bxy + cy^2$ be a binary quadratic form with integer coefficients and non-zero discriminant $\Delta(f)$. It is well-known that $f$ is $\operatorname{GL}_2(\mathbb{R})$-equivalent (via substitution) to either $x^2 + y^2$ or $xy$, depending on the sign of $\Delta(f)$. Is there an analogue for this statement when the local field $\mathbb{R}$ is replaced with $\mathbb{Q}_p$ for some finite prime $p$, or the $p$-adic integer $\mathbb{Z}_p$? That is, does there exist for each prime $p$ a positive integer $k_p$ such that every binary quadratic form $f$ with integer coefficients is $\operatorname{GL}_2(\mathbb{Z}_p)$-equivalent to one of $k_p$ binary quadratic forms with integer coefficients?

Let $f(x,y) = ax^2 + bxy + cy^2$ be a binary quadratic form with integer coefficients and non-zero discriminant $\Delta(f)$. It is well-known that $f$ is $\operatorname{GL}_2(\mathbb{R})$-equivalent (via substitution) to either $x^2 + y^2$ or $xy$, depending on the sign of $\Delta(f)$. Is there an analogue for this statement when the local field $\mathbb{R}$ is replaced with $\mathbb{Q}_p$ for some finite prime $p$, or the $p$-adic integers $\mathbb{Z}_p$? That is, does there exist for each prime $p$ a positive integer $k_p$ such that every binary quadratic form $f$ with integer coefficients is $\operatorname{GL}_2(\mathbb{Z}_p)$-equivalent to one of $k_p$ binary quadratic forms with integer coefficients?