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 $m\ll_{k,\epsilon} d^{2+\epsilon}$ for any $\epsilon >0$, 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 $m\ll_{k,\epsilon} \left(d \sqrt {d+g}\right)^{1+\epsilon}$. 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.