nt.number theory – Norm forms, slicing, and ideal classes


Let $K$ be a number field, which we may suppose satisfies $n = (K : mathbb{Q}) geq 3$. Let $mathcal{O}_K$ be the ring of integers of $K$, and let ${omega_1, cdots, omega_{n}}$ be a basis of $mathcal{O}_K$. Define the norm form of $K$ to be the homogeneous polynomial defined by

$displaystyle N_{K/mathbb{Q}}(mathbf{x}) = prod_{sigma : K hookrightarrow mathbb{C}} left(sigma(omega_1) x_1 + cdots + sigma(omega_n) x_nright),$

where the product runs over embeddings $sigma: K hookrightarrow mathbb{C}$.

We can extend this definition of norm form to any order $mathcal{O}$ contained in $mathcal{O}_K$, by replacing the basis ${omega_1, cdots, omega_n}$ with a basis for $mathcal{O}$.

Now suppose that $mathcal{O}$ is an order, with basis ${nu_1, cdots, nu_n}$, say, and let $N_{mathcal{O}}(mathbf{x})$ be the corresponding norm form. Let us take an element $A in text{GL}_n(mathbb{Z})$, and consider $G : = N_{mathcal{O}}(A mathbf{x})$. Now restrict $N_{mathcal{O}}, G$ to the 2-dimensional space defined by $x_3 = cdots = x_n = 0$, to obtain binary $n$-ic forms $f(x_1, x_2),g(x_1, x_2)$ say.

What can we say about $f,g$ in general? In particular, are their discriminants connected in any way? What about the rings/ideal classes they represent?