Geodesic
Rational torsion points on abelian surfaces with quaternionic multiplication
Forum of Mathematics, Sigma, Tome 12 (2024)

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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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     author = {Jef Laga and Ciaran Schembri and Ari Shnidman and John Voight},
     title = {Rational torsion points on abelian surfaces with quaternionic multiplication},
     journal = {Forum of Mathematics, Sigma},
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     year = {2024},
     doi = {10.1017/fms.2024.105},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.105/}
}
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Jef Laga; Ciaran Schembri; Ari Shnidman; John Voight. Rational torsion points on abelian surfaces with quaternionic multiplication. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.105

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