On the semistability of binary forms over number fields

TL;DR

This paper explicitly characterizes when binary forms over number fields are semistable locally and globally, providing formulas for the associated moduli heights for forms of degrees 4, 6, 8, and 10.

Contribution

It offers explicit constructions of semistable forms over local and global fields and computes their moduli heights for specific degrees, advancing understanding of binary form stability.

Findings

Explicit formulas for semistable forms over local fields.

Global semistability criteria for binary forms.

Computed moduli heights for degrees 4, 6, 8, and 10.

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