Reference request: Quotient of quadratic forms

Question

Is there a way to determine if there exist integers x and y that, when plugged into two quadratic forms $P(x,y)$ and $Q(x,y)$, their quotient will also be an integer (i.e., $P \equiv 0 \pmod{Q}$)? I've found a reference that establishes a Hasse principle for the division of quadratic forms, but I haven't found any further sources or examples.

Answer 1

What you want is the difference $P - nQ$ to have a square discriminant for some integer $n.$ Then it is isotropic, there is a (primitive) pair $(x,y)$ that makes it zero, because it factors as the product of two linear forms.

If $P(x,y) = a x^2 + b xy + c y^2$ while $Q(x,y) = d x^2 + e xy + f y^2,$ the discriminant is $$ \Delta = (b - ne)^2 - 4(a - nd)(c - nf) $$ or $$ \Delta = (e^2 - 4 df) n^2 + (4af + 4cd -2be)n +(b^2 -4ac).$$ And you want an $n$ for which the thing is square (including zero) as in

$$ m^2 = (e^2 - 4 df) n^2 + (4af + 4cd -2be)n +(b^2 -4ac).$$

This is not going to be possible if the right hand side is always negative, which might happen if both $P,Q$ are positive definite and the mixed term is of modest absolute value.

Worth trying some example pairs $P,Q$