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arXiv · 2308.15193

Rational torsion points on abelian surfaces with quaternionic multiplication

Jef Laga·Ciaran Schembri·Ari Shnidman·John Voight
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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 $A(\mathbb{Q})$.

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Jef Laga, Ciaran Schembri, Ari Shnidman, John Voight. 2023-08-29. Rational torsion points on abelian surfaces with quaternionic multiplication. https://doi.org/10.1017/fms.2024.105

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