Divisibility of torsion subgroups of abelian surfaces over number fields

Abstract

Let $A$ be a 2-dimensional abelian variety defined over a number field $K$. Fix a prime number $\ell$ and suppose $\#A({\mathbb{F}}_p)\equiv 0(\mathrm{mod}{\ell }^2)$ for a set of primes $\mathfrak{p}\subset {\mathcal{O}}_K$ of density 1. When $\ell =2$ Serre has shown that there does not necessarily exist a $K$-isogenous $A^{\prime }$ such that $\#A^{\prime }(K{)}_{\mathrm{tors}}\equiv 0(\mathrm{mod}4)$. We extend those results to all odd $\ell$ and classify the abelian varieties that fail this divisibility principle for torsion in terms of the image of the mod-${\ell }^2$ representation.

31 pages, 5 sections. Numerous changes in exposition have been made since the last version and section 6 has been removed, but all main results are the same. This is the version that will appear in the Canadian Journal of Mathematics

References in corpus (1)

• A database of genus 2 curves over the rational numbers