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 $A(\mathbb{Q})$.

37 pages, comments welcome

References in corpus (2)

• Computing classical modular forms
• The geometry and arithmetic of bielliptic Picard curves