Let $K = \mathbb{Q}(\sqrt{d})$ be a quadratic number field, and let $\mathcal{O}_K$ be its ring of integers, and let $D$ be the discriminant of $\mathcal{O}_K$. It is a well-known result (see for example these notes of Andrew Granville: http://www.dms.umontreal.ca/~andrew/Courses/Chapter4.pdf) that an integer $m$ is the norm of an ideal in $\mathcal{O}_K$ if and only if

(1) $$\displaystyle \sum_{n | m} \left(\frac{D}{n}\right) \geq 1.$$

In fact, the left hand side of the above is precisely the number of representation of $m$ by integral binary quadratic forms of discriminant $D$.

Is there an analogue of this result for cubic norms? Here, the appropriate substitute for $D$ ought to he a $\operatorname{GL}_2(\mathbb{Z})$-equivalence class of integral binary cubic forms; this is because discriminants $D$ parametrize quadratic orders, while cubic orders are parametrized by $\operatorname{GL}_2(\mathbb{Z})$-classes of binary cubic forms. My question is thus as follows:

Let $K$ be a cubic field, and let $\mathcal{O}_K$ be its ring of integers. Let $F$ be a representative of the $\operatorname{GL}_2(\mathbb{Z})$-class of integral binary cubic forms which represents $\mathcal{O}_K$. Is there a criterion that depends solely on $F$, analogous to (1), which determines whether a given integer $m$ is the norm of an ideal in $\mathcal{O}_K$?


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