# Which integers are represented by fourth powers (with respect to Gauss composition) of binary quadratic forms?

Let $$m$$ be a positive integer. We say that $$m$$ can be represented by a binary quadratic form

$$\displaystyle f(x,y) = ax^2 + bxy + cy^2, a,b,c \in \mathbb{Z}$$

if there exist integers $$u,v$$ such that $$m = f(u,v)$$. We say that $$m$$ is primitively representable if $$u,v$$ can be chosen so that $$\gcd(u,v) = 1$$. Note that both notions are preserved under $$\text{SL}_2(\mathbb{Z})$$-equivalence, so we may say that $$m$$ is representable by the class $$[f]$$ of $$f$$.

Recall that for a fixed integer $$D$$, the set of $$\text{SL}_2(\mathbb{Z})$$-equivalence classes of binary quadratic forms of discriminant $$D$$ forms a group: this was proved by Gauss, and this phenomenon is now known as the Gauss composition laws. Denote by $$H(D)$$ the form class group of discriminant $$D$$ with respect to Gauss composition.

Suppose that $$f$$, or rather the equivalence class $$[f]$$, is a $$4$$th power in the form class group $$H(D)$$ of discriminant $$D$$: that is, there exists a form $$g$$ of discriminant $$D$$ such that $$[g]^4 = [f]$$, where the operation is Gauss composition. Let $$H_4(D)$$ denote the subgroup of $$H(D)$$ consisting of 4-th powers.

Is it known how to determine whether an integer $$m$$ belongs to the following set?

$$\displaystyle \{n \in \mathbb{Z} : \exists [f] \in H_4(D) \text{ such that } n \text{ is primitively representable by } [f]\}?$$

Using genus theory, the question with 4th powers replaced with squares can be answered as follows. An equivalence class $$[f] \in H(D)$$ is a square, i.e., there exists $$[g] \in H(D)$$ such that $$[f] = [g]^2$$, if and only if $$[f]$$ is in the principal genus. Using this, one shows that $$m$$ is representable by a square in $$H(D)$$ if and only if the sum

$$\displaystyle \sum_{c | D} \left(\frac{m}{c}\right) \ne 0.$$

Edit: By the following paper provided by Will Jagy ( zakuski.math.utsa.edu/~kap/Estes_Pall_1973.pdf), we have $$[f] \in H_4(D)$$ if and only if $$[f]$$ is in the principal spinor genus; see https://en.wikipedia.org/wiki/Spinor_genus

• If I recall correctly, the analogous question for the subgroup of squares is answered by genus theory: $m$ is represented by the square of a form of discriminant $D<0$ if and only if the Legendre symbol $(\frac{m}{p})=1$ for all odd $p|D$, plus an analogous modulo 4 or 8 if $D$ is even. (Caveat, maybe more complicated if $D$ is not fundamental?) – Stopple Aug 14 '19 at 17:51
• @Stopple that's correct. Do you think it's helpful to include this fact in the body of the question? – Stanley Yao Xiao Aug 14 '19 at 18:21
• For me, more historical context is helpful – Stopple Aug 14 '19 at 18:29