On the maximal order of a torsion point on a curve in ${\mathbb{G}}_m^n$

Abstract

Let $\mathcal{C}$ be an irreducible algebraic curve in ${\mathbb{G}}_{\mathrm{m}}^2$; we are concerned with the maximal order $m=m(\mathcal{C})$ of a torsion point on $\mathcal{C}$. We suppose that $\mathcal{C}$ is defined over a number field $k$, that it is not a translate of an algebraic subgroup by a torsion point, and we denote by $d$ its degree and by $g$ its genus. It is known that $m{\ll }_{k,ϵ}{d}^{2+ϵ}$ for any $ϵ>0$, which, as shown below, is nearly best-possible if only the degree is taken into account. Here, by means of a new method, we prove an upper bound (actually for curves in ${\mathbb{G}}_{\mathrm{m}}^n$) which implies in particular $m{\ll }_{k,ϵ}{\left(d\sqrt{d+g}\right)}^{1+ϵ}$. This gives back the above result and for small $g$ it improves on it. The appearance of the genus seems to be a new feature in this kind of problems.