JournalsrlmVol. 19 , No. 1DOI 10.4171/rlm/509

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

Abstract

Let \C\C be an irreducible algebraic curve in \Gm2\G_{\rm{m}}^2; we are concerned with the maximal order m=m(\C)m=m(\C) of a torsion point on \C\C. We suppose that \C\C is defined over a number field kk, that it is not a translate of an algebraic subgroup by a torsion point, and we denote by dd its degree and by gg its genus. It is known that mk,ϵd2+ϵm\ll_{k,\epsilon} d^{2+\epsilon} for any ϵ>0\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 \Gmn\G_{\rm{m}}^n) which implies in particular mk,ϵ(dd+g)1+ϵm\ll_{k,\epsilon} \left(d \sqrt {d+g}\right)^{1+\epsilon}. This gives back the above result and for small gg it improves on it. The appearance of the genus seems to be a new feature in this kind of problems.