Large rational torsion on abelian varieties

A method of searching for large rational torsion on Abelian varieties is described. A few explicit applications of this method over Q give rational 11- and 13-torsion in dimension 2, and rational 29-torsion in dimension 4

Sequences of rational torsions on abelian varieties

We address the question of how fast the available rational torsion on abelian varieties over Q increases with dimension. The emphasis will be on the derivation of sequences of torsion divisors on hyperelliptic curves. Work of Hellegouarch and Lozach (and Klein) may be made explicit to provide sequences of curves with rational torsion divisors of orders increasing linearly with respect to genus. The main results are applications of a new technique which provide sequences of hyperelliptic curves for all torsions in an interval $[a_g, a_g+b_g]$ where $a_g$ is quadratic in g and $b_g$ is linear in g. As well as providing an improvement from linear to quadratic, these results provide a wide selection of torsion orders for potential use by those involved in computer integration. We conclude by considering possible techniques for divisors of non-hyperelliptic curves, and for general abelian varieties

Courbes elliptiques semi-stables et corps quadratiques

RésuméSoientKun corps quadratique,Kune clôture algébrique deKetEune courbe elliptique semi-stable définie surK. On se propose de démontrer que l'opération du groupe de Galois Gal(K/K) sur les points dep-torsion deEest irréductible, sipest un nombre premier plus grand qu'une constante qui ne dépend que deK