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 $d=4,6,8,10$.

References in corpus (2)

• From hyperelliptic to superelliptic curves
• Positivity of GIT heights of zero-cycles and hyperplane arrangements