Question about the definition of representability of a quadratic form

Question

Say I have an integral quadratic form $q$ on $\mathbb{Z}^r$ and another integral binary quadratic form $\tilde{q}$. What does it mean for $q$ to primitively represent $\tilde{q}$? I can't seem to find the definition anywhere. Of course $q$ primitively represents an integer $m$ if there exists a primitive element $\alpha\in\mathbb{Z}^r$ such that $q(\alpha)=m$, but I'm not sure what it means to represent another quadratic form (especially when the other quadratic form is not defined on the same amount of variables!).

Answer 1

You take the Gram matrix, which is generally taken to be the Hessian matrix of second partial derivatives. Let $A$ be the Gram matrix for your $q,$ and $B$ be the Gram matrix for the other one. Representation is a rectangular matrix $P$ such that $$ P^T A P = B. $$ Primitive representation is if the GCD of all the entries of $P$ is $1.$ Where are you reading about this?

The case with which you are familiar is when $P$ is a column vector and $B$ is a 1 by 1 matrix, that is, a number.

On page 18, remark 35, you need Estes and Pall (1973), which I put at TERNARY. The principal forms are the squares in the class group. The spinor kernel is the fourth powers in the group. Your $\bar{s}(D)$ is the number of spinor genera per genus, which is the same as the number of squares divided by the number of fourth powers.