# On the maximal order of a torsion point on a curve in <em><strong>G</strong><sup>n</sup><sub>m</sub></em>

### Umberto Zannier

Scuola Normale Superiore, Pisa, Italy### Pietro Corvaja

Università di Udine, Italy

## 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.

## Cite this article

Umberto Zannier, Pietro Corvaja, On the maximal order of a torsion point on a curve in <em><strong>G</strong><sup>n</sup><sub>m</sub></em>. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19 (2008), no. 1, pp. 73–78

DOI 10.4171/RLM/509