Rational torsion points on abelian surfaces with quaternionic multiplication

TL;DR

This paper investigates the structure of rational torsion points on abelian surfaces with quaternionic multiplication, establishing bounds and classifications similar to Mazur's theorem for elliptic curves.

Contribution

It provides bounds on torsion subgroup sizes and a complete classification for abelian surfaces with quaternionic multiplication, extending Mazur's theorem.

Findings

Torsion subgroup of such abelian surfaces is at most 18 in order.

The torsion subgroup is exactly 12-torsion.

Complete classification of torsion subgroups for GL2-type abelian surfaces.

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 $A(\mathbb{Q})$.