Does the set of possible values of a binary quadratic form determine the form

Question

If two forms have the same range and discriminant, then due to reduction to a unique reduced form(that depends only on the 2 smallest numbers in the range of the form), we can conclude that the two forms have to be equivalent.

Is this still true if the two forms have different discriminants. That is, can there be two forms of different discriminants that have the same range?

Edit: I am only mainly interested in negative discriminants

Answer 1

Well, $x^2 + xy + y^2$ and $x^2 + 3 y^2$ represent exactly the same numbers. The main thing for (positive) binary forms is determining the primes represented. There are a finite number of pairs of forms (and some infinite families), of differing discriminants, that represent the same primes. Kaplansky and I wrote up a provisional list. the project was finished, and published, by John Voight, who was a graduate student at the time. He is now at Dartmouth. Here we go, item number 6 at https://math.dartmouth.edu/~jvoight/research.html As this is a necessary condition for representing the same numbers, you can just check his list to find all occurrences of your stronger condition.

You may note, in his tables, the frequency of discriminants being two or four times the lowest on in that section. That is to be expected: given $f(x,y),$ taking $f(u-v, u+v)$ and $f(u, 2v)$ (or perhaps $f(2u,v)$) automatically gives forms that represent a subset of the numbers represented by $f$ itself. The great majority of the time, a strict subset, but not always. A useful example is that $$ x^2 + xy + 2k y^2 $$ and $$ x^2 + (8k-1) y^2 $$ represent exactly the same ODD numbers, and therefore the same primes as soon as $k > 1.$ Among even numbers, the first one represents both $2k$ and $2k+2,$ the second one does not represent both.

Meanwhile, you are not wrong about taking the three or four smallest values and having that determine the form. You are allowed to demand the form to be reduced.

Answer 2

I think I have a possible answer:

Let $f(x,y) = ax^2 _ bxy + cy^2$ be any reduced form with negative discriminant and $a<c$. Then $f(x,y) \geq (a - |b| + c)min(x^2,y^2)$. Therefore the 3 smallest values in the range of the function will be $a,c,a-|b|+c$ in that order. This defines the form upto equivalence.

I do not really know how to differentiate between the case where $a < c$ and $a = c$ though.