Given two binary quadratic forms f and g which have the same determinant and represent the same prime p, show that f and g are equivalent (related by a 2*2 matrix of determinant 1 or -1).
My approach: Obviously f and g properly represent p. Then f ~ (p,k,l) and g~(p,k',l'). Now it remains to show that (p, k, l) and (p, k', l') are related by a 2*2 matrix with det = 1 or -1. (thus not necessarily properly equivalent; general equivalent suffices)
Any hint will be appreciated! Thanks!
Lets say the forms are $(p,k,l)$ and $(p,n,m)$. Then $$k^2-4pl=n^2-4pm$$ and thus (since you dont care about improper equivalence) we may assume $$k\equiv n\mod 2p$$
As is well known the transform $$x\mapsto x+by$$ $$y\mapsto y$$ leaves the initial coefficient $p$ unchanged and changes the middle coefficient by $+2bp$. We are done.