4
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Do you have any evidence to support the conjecture posed in Question 1? $\endgroup$ Apr 14 at 18:05
  • 1
    $\begingroup$ I was careful to call it a "Question", not a conjecture. I don't have evidence for it. $\endgroup$
    – Joseph
    Apr 14 at 18:21

1 Answer 1

5
$\begingroup$

No, it is not true that $M(D) = o(|D|^{3/2})$. For example, if $D = -4abcd$ where $a,b,c,d$ are odd, pairwise coprime and of roughly equal size, then the forms $ab\,x^2+cd\,y^2, ac\,x^2+bd\,y^2, ad\,x^2+bc\,y^2$ represent 2-torsion classes $J_1,J_2,J_3$ in the class group whose product is the identity, and each has $H(J_i) \sim D^{1/2}$ so $H(J_1)H(J_2)H(J_3) \sim D^{3/2}$. Indeed it seems more plausible (though probably not easy to prove) that the lim inf of $M(D)/D^{3/2}$ is $C^3$.

$\endgroup$
1
  • $\begingroup$ Thank you! This is quite helpful. $\endgroup$
    – Joseph
    Apr 15 at 5:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.