Suppose $K$ is a quadratic imaginary field with discriminant $D$; let $S$ denote the ring of integers in $K$. For a fractional $S$-ideal $J$, define the **height** of $J$, denoted $H(J)$, to be the minimum of the norms of nonzero elements of $J$, normalized by the norm of $J$:

$H(J) = min_{\alpha \in J, \alpha \neq 0} \frac{N_{K/\mathbf{Q}}(\alpha)}{N(J)}.$

Note that $H(J)$ descends to the class group. If $f_J(x,y)$ is the binary quadratic form associated to $J$ by Gauss composition, then $H(J)$ is the smallest positive integer represented by the binary quadratic form $f_J(x,y)$. Indeed, if $u,v$ are a $\mathbf{Z}$-basis of $J$, then $f_J(x,y) = \frac{N(ux+vy)}{N(J)}$. By either reduction theory for binary quadratic forms, or the Minkowski bound, one has that there exists a constant $C$, independent of $D$, so that $H(J) \leq C |D|^{1/2}$ for all $J$.

Now suppose $(J_1, J_2, J_3)$ is a triple of ideal classes for $K$ with product equal to $1$ in the class group: $[J_1] [J_2] [J_3] = 1$ in $Cl(K)$. Such triples arise in Bhargava's paper "Higher Composition Laws I". Trivially, one has $H(J_1)H(J_2)H(J_3) \leq C^3 |D|^{3/2}$.

**Question 1**: Can one improve the above bound on the product of the heights, even slightly?

More precisely, for each $D$, define $M(D)$ to be maximum of such products of heights:

$M(D) = max_{[J_1], [J_2], [J_3] \text{ in } Cl(K) \text{ with } [J_1] [J_2] [J_3] = 1} H(J_1)H(J_2)H(J_3)$.

**Question 1 (precise verion)**: Does $lim_{|D| \rightarrow \infty} \frac{M(D)}{|D|^{3/2}}$ exist and equal $0$?

A positive answer to Question 1 can be thought of as some sort of refined reduction theory for Bhargava cubes, hence the title of the question.

One special case of this question is to restrict to the $3$-torsion in class groups: Define $E_3(D) = max_{[J] \in Cl(K): [J]^3 =1}H(J)$.

**Question 2** Does $lim_{|D|\rightarrow \infty} E_3(D) |D|^{-1/2}$ exist and equal $0$?

I don't know the answer to Question 2 either.