Rational torsion points on abelian surfaces with quaternionic multiplication
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...
| Published in: | Forum of Mathematics, Sigma |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2024-01-01
|
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001051/type/journal_article |
| Summary: | 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
$A({\mathbb {Q}})$
. |
|---|---|
| ISSN: | 2050-5094 |