Rational torsion points on abelian surfaces with quaternionic multiplication

Autor: Jef Laga, Ciaran Schembri, Ari Shnidman, John Voight
Jazyk: angličtina
Rok vydání: 2024
Předmět:
Zdroj: Forum of Mathematics, Sigma, Vol 12 (2024)
Druh dokumentu: article
ISSN: 2050-5094
DOI: 10.1017/fms.2024.105
Popis: 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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