On the maximal order of a torsion point on a curve in <em><strong>G</strong>ⁿ_m</em>

Abstract

Let \( \C \) be an irreducible algebraic curve in \( \G_{\rm{m}}^2 \); we are concerned with the maximal order \( m=m(\C) \) of a torsion point on \( \C \). We suppose that \( \C \) 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 \( \G_{\rm{m}}^n \)) 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.