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In the recent paper by Bhargava and Harron, arXiv:1309.2025. They prove that cubic, quartic and quintic fields have equidistributed "shapes" as discriminant gets large.

Their notion of shape simultaneously considers all embeddings of the ring of integers into C and R.

$$ q(x) = \sigma_1(x)^2 + \dots + \sigma_r(x)^2 + 2|\tau_1(x)|^2 + \dots + |\tau_s(x)|^2$$

Their paper only goes up to degree 5, but I will content myself with degree 2 and 3. How can we compute $q(x)$ in a few examples ?

  • $x^2 - x - 1$ two real roots
  • $x^2 + x + 1$ two complex roots
  • $x^3 - x - 1$ two complex, one real root

I found some prior discussions on cubic rings of integers. I think the quadratic form discussed here is a bit different


In addition Bhargava and Harron have this to say about "shapes"

The shape of $\mathcal{O}_K$ is defined to be the (n − 1)-ary quadratic form, up to scaling by $\mathbb{R}^×$, obtained by restricting $q$ to $\{x ∈ \mathbb{Z} + n\mathcal{O}_K : Tr_{K/ Q} (x) \geq 0\}$, which is therefore well-defined up to the action of $\mathbb{G}_m(\mathbb{R}) × GL_{n−1}(\mathbb{Z})$.

Alternatively, the shape of OK may be defined as the (n − 1)-ary quadratic form, up to scaling by $\mathbb{R}^×$, obtained by restricting the rational quadratic form $q$ on $K$ to the projection of $\mathcal{O}_K$ onto the hyperplane in K that is orthogonal to 1.

Hence the shape may be viewed as an element of $$\mathcal{S}_{n−1} := GL_{n−1}(\mathbb{Z})\backslash GL_{n−1}(\mathbb{R})/ GO_{n−1}(\mathbb{R})$$ which we call the space of shapes of lattices of rank n − 1.

I have trouble understanding what this "restricted" quadratic form should look like.

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