Is the set of integers represented by a quadratic form of non-fundamental discriminant a subset of the rep. set of a form of fundamental discriminant?

Question

I am currently working with positive-definite, reduced, primitive, integral binary quadratic forms, and I have noticed something interesting.

Conjecture: Let $Q$ be a form of non-fundamental discriminant $\Delta$. Let $K=\mathbb{Q}(\sqrt{\Delta})$ be the associated quadratic field of discriminant $\Delta_K$ (which we note is fundamental). Let $R=\{m: m=Q(x,y), x,y\in\mathbb{Z}\}$, the set of integers represented by $Q$. Then $R\subseteq R_K$, where $R_K=\{m: m=Q_K(x,y), x,y\in\mathbb{Z}\}$ for a fixed form $Q_K$ of fundamental discriminant $\Delta_K$.

Example: Let $Q(x,y)=x^2+8y^2$. Then $\Delta=-32$. Let $R=\{m: m=Q(x,y), x,y\in\mathbb{Z}\}$. Then $R\subseteq\{m: m=x^2+2y^2, x,y\in\mathbb{Z}\}$.

This seems like a very basic fact about fundamental and non-fundamental discriminants, but I cannot find it anywhere in the literature. The only thing that has been suggested to me is the term maximal quadratic form, but I can't find that either. Is this a known theorem?

Answer 1

This is true, of course. The book you want is Binary Quadratic Forms by D. A. Buell. He talks about how, given some discriminant $\Delta$ and class number(primitive forms) $h(\Delta),$ we can predict $h(4 \Delta)$ and $h(p^2 \Delta).$ Pages 117-118, let me check.

I did find slides for a 2010 talk by Buell. It is not about your question, however the style is quite reminiscent of his book.

In the reverse direction (Kaplansky used to do these two variables at a time, fine as you have just two) we can, given discriminant $\Delta p^2,$ we can find a form $SL_2 \mathbb Z$ equivalent to the original that is now of the form (with $a$ not divisible by $p$) $$ \langle a,bp,cp^2 \rangle \; \; , $$ which descends to $$ \langle a,b,c \rangle \; \; . $$

Let's see, if $$ f(x,y) = a x^2 + b p x y + c p^2 y^2 $$ and $$ g(x,y) = a x^2 + b x y + c y^2, $$ then $$ f(x,y) = g(x,py) \; , $$ making your subset idea explicit.

This is page 119 in Buell, with a repeat of the comment that the one-to-many maps he gives from discriminant $\Delta$ upwards to $\Delta p^2$ is surjective.

In short, Chapter 7, section 1, pages 109-119 in Buell. The section title is "Nonfundamental Discriminants."

I see, students of Pete L. Clark not long ago: https://arxiv.org/abs/1708.04877