On the semistability of binary forms over number fields

Abstract

Let $K$ be a number field, ${\mathcal O}_K$ its ring of integers, and $f(x, y) \in {\mathcal O}_K[x, y]$ a binary form with integer coefficents. For any given prime $p \in {\mathcal O}_K$ we determine explicitly a binary form $g$ (resp. $\bar f$), $\mbox{GL}_2 (K)$-equivalent to $f$ which is semistable over the local field $K_p$ (resp. the global field $K$). Moreover, if $\xi(f) $ is the corresponding moduli point in the weighted projective space ${\mathbb{WP}}_{\mathbf w}^n (K)$ for a strictly semistable binary form $f$, we determine the weighted moduli height $\mathfrak{h} (\xi(f))$ for $d=4, 6, 8, 10$.