When is the set of numbers represented by certain quaternary quadratic forms completely multiplicative?

2012-02-20T22:23:32Z

Expired by this question http://mathoverflow.net/questions/88905/a-quadratic-form-represents-all-primes-except-for-the-primes-2-and-11 I would like to know some simple sufficient conditions for when the set of numbers integrally represented by a positive four-variable quadratic form is completely multiplicative, with $$ h(x,y,z,w) = f(x,y) + B g(z,w), $$ while $f,g$ are positive primitive binary quadratic forms of the same discriminant, and integer $B \geq 1.$ The only example I am completely sure about is $$ h(x,y,z,w) = x^2 + A y^2 + B z^2 + A B w^2. $$ The reason this one is easy is that we can construct quaternions $$ x + y i \sqrt A + z j \sqrt B + w k \sqrt {A B}, $$ so the quadratic form is the norm and multiplication is built in. In particular, taking $B=1,$ we have the principal form repeated.

Noam Elkies pointed out that the condition of complete multiplicativity does not hold for repeated binary $3 x^2 + 7 y^2,$ as $$ h(x,y,z,w) = 3 x^2 + 7 y^2 + 3 z^2 + 7 w^2$$ represents $3$ and $7$ but not $21.$

I had this feeling that the fact the $3 x^2 + 2 x y + 4 y^2$ was in a discriminant of one genus and class number three would suffice for the form in question 88905. By itself, the binary form represents a completely multiplicative set, as its cube in the class group is the identity but its square is simply its own opposite class. But I am not entirely sure about the quaternary in question 88905. Furthermore, I am getting other examples, class number five and so on.

So, that is the question, simple sufficient conditions for complete multiplicativity of integers represented by $$ h(x,y,z,w) = f(x,y) + B g(z,w).$$

EDIT, 2:56 pm: Note that the numbers represented by this $f(x,y) + B g(z,w)$ can always be multiplied by any number represented by the principal form of the same discriminant, as this holds true separately for $f,g.$ For this reason, if the form in question 88905 is completely muliplicative (as seems very likely) it suffices to consider primes $p \neq 2,11$ along with $2p$ for $p \neq 2,11.$ Noam has already done such the first case.

EDIT TOOOO, 5:06 pm: Noam says that, in question 88905, his answer extends to show that the set of numbers represented is indeed multiplicative, but he does not expect there to be a direct proof of that. So one might say that I am asking for situations where there is a direct proof. For example, there are plenty of forms in four variables that represent all positive integers (if they are integer-matrix, they just need to represent the numbers up to 15, if not integer matrix then we need them to represent up to 290 to be sure). Anyway, a positive form that represents all numbers integrally is "completely multiplicative" in the way I define it, but that does not mean there is any nicer proof.

EDIT THREE, 9:18 pm: I have been running examples. Another counterexample is repeating the binary $3 x^2 + x y + 3 y^2,$ of discriminant $-35,$ the quaternary represents $3,7$ but not $21.$ However, the class number of the principal genus does not seem to matter. So a reasonable revised question is, how about when both $f,g$ are in the principal genus? NO, not good enough, same results with $3 x^2 + 3x y + 4 y^2,$ of discriminant $-39,$ principal genus.

I have no idea what is going on. Many of these clearly work, some just don't. If they fail, it seems to be with very small numbers.

When is the set of numbers represented by certain quaternary quadratic forms completely multiplicative? - MathOverflow

When is the set of numbers represented by certain quaternary quadratic forms completely multiplicative?