Rational torsion points on abelian surfaces with quaternionic multiplication

Abstract

Let A be an abelian surface over ${\mathbb {Q}}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur’s theorem for elliptic curves, we show that the torsion subgroup of $A({\mathbb {Q}})$ is $12$ -torsion and has order at most $18$ . Under the additional assumption that A is of $ {\mathrm{GL}}_2$ -type, we give a complete classification of the possible torsion subgroups of $A({\mathbb {Q}})$ .

Keywords

• 14K15
• 14G05
• 11G10