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I have heard that Gauss studied integer binary quadratic forms with the middle term divisible by two $$f(x,y) = a x^2 + 2bxy + c y^2. $$ However, more modern treatment does not do this. Buell's "Binary Quadratic Forms" starts with the line:

We consider here the binary quadratic forms in two variables $f(x,y) = a x^2 + bxy + cy^2$ of discriminant $b^2 - 4ac = \Delta$.

But Gauss must have handled the odd terms somehow, otherwise there would be no forms with a discriminant equal to (+/-) an odd prime. And when it comes time to discuss composition of quadratic forms, I'd think it would cause havoc with missing elements in the set of forms of a given discriminant.

Similarly, in discussions of more modern work, the binary quadratic form $$f(x,y) = a x^2 + 2bxy + c y^2$$ and binary cubic form $$f(x,y) = a x^3 + 3bx^2y + 3cxy^2 + d y^3$$ seem to be important. For example, in Bhargava's "Higher composition laws I", Bhargava cubes can be related to cubic forms if the middle terms are a multiple of 3, and to pairs of the quadratic forms if the middle terms are even.

So what is special about the "form" of these forms?
Does this continue, and binary quadratic forms are more convenient to study if they look like this? $$f(x,y) = a x^4 + 4b x^3 y + 6c x^2 y^2 + 4d x y^3 + e y^4$$

And, more explicitly, in the case of the binary quadratic forms:
What advantage did an even middle term have that made Gauss study it that way?
And how does the modern study (where we do allow odd middle terms) relate to his work?
Does this also apply for cubic and higher binary forms?

Apologies for the soup of questions, but these extra factors of Binomial coefficients are confusing me. Especially when it sounds like there is some simple reason why these are equivalent ways of studying these objects. Unfortunately, this simple reason is not obvious to me.

(For reference, I am a curious undergraduate, probably reading stuff before I should ... so no, I do not understand everything in Buell's book or Bhargava's paper, but I'd very much like to learn more.)

Note: This could be viewed as an extension of this question:
What's so special about the form $ax^2+2bxy+cy^2$?
Although the answers seems focused on the generality part of the question rather than the issue about the coefficient being "even".

dwbarkley
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  • Gauss used b^2-ac as his discriminant. And because he worked with imprimitive forms (all coefficients have a nontrivial common factor) he could consider a form f with odd middle coefficient just by working with 2f. For the purposes of reduction and composition, everything just ends up multiplied by 2. – Barry Smith Feb 23 '20 at 04:21
  • @BarrySmith I'm not sure I understand. If he had a quadratic form with an odd middle coefficient, then "working with 2f" would change the discriminant, even by that different definition of the discriminant. For example, the set of bqf with (usual def) discriminant -53 has two classes of forms, all with odd middle coefficient. Working with "2f" now we are looking at bqf with discriminant -4*53, which there are now six classes of forms. Would he consider the later to have eight? – dwbarkley Feb 23 '20 at 04:52
  • Do you mean -51? There are 8 reduced forms of discriminant -4*51, of which six are primitive and two are imprimitive. Those two are just scaled versions of the two primitive forms of discrimiantn -51. Gauss defined composition of forms of different discriminants even, but noted how certain things are nicer when combining forms of the same discriminant. His composition can also use imprimitive forms. You can take the two imprimitive forms and compose them with each other or themselves and get a multiple of what you get if you instead composed the corresponding primitive forms. – Barry Smith Feb 23 '20 at 21:20
  • @BarrySmith Oops, not sure how I mistyped 51 as 53 twice. Kudos for being able to still understand anyway. – dwbarkley Feb 24 '20 at 02:53
  • I did not notice until now, but pari-gp seems to let me compose a primitive and imprimitive form. I was curious what the structure would look like for the set of 8 reduced forms of discriminant -451, including the imprimitive forms. Composing with an imprimitive form ruins the group structure: Qfb(5,4,11)Qfb(6,6,10) = Qfb(3,0,17)Qfb(6,6,10) = Qfb(5,-4,11)Qfb(6,6,10) = Qfb(2, 2, 26). I did not fully appreciate until now how carefully the modern definitions disentangled all this. It must have been very confusing to see some structure and try to figure out how to extract it cleanly. – dwbarkley Feb 24 '20 at 03:04

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