Equivalence of binary quadratic forms over $\operatorname{GL}_2(\mathbb{Q}_p)$ or $\operatorname{GL}_2(\mathbb{Z}_p)$

Question

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?

Answer 1

The number of classes of (regular) binary quadratic forms over $\mathbb{Q}_p$ equals $7$ when $p\neq 2$, and it equals $15$ if $p=2$. See Section IV.2.3 of Serre: A course in arithmetic.

The number of classes of binary quadratic forms over $\mathbb{Z}_p$ is naturally infinite, but they can be represented by explicit canonical forms. See Sections 8.3-8.4 in Cassels: Rational quadratic forms.

Probably this does not fully answer your question as you want to work with integral coefficients only, but the above references will help you out. In particular, each class over $\mathbb{Q}_p$ can be represented by a form over $\mathbb{Z}_p$.