The geometry and arithmetic of bielliptic Picard curves
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916513541259264 |
|---|---|
| author | Laga, Jef Shnidman, Ari |
| author_facet | Laga, Jef Shnidman, Ari |
| contents | We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2 + b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem in this context and give a geometric proof of the fact that $P$ has quaternionic multiplication (QM) by the quaternion order of discriminant $6$. This allows us to describe the Galois action on the geometric endomorphism algebra of $P$. As an application, we classify the torsion subgroups of the Mordell-Weil groups $P(\mathbb{Q})$, as both abelian groups and $\text{End}(P)$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15297 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The geometry and arithmetic of bielliptic Picard curves Laga, Jef Shnidman, Ari Algebraic Geometry Number Theory 14H45 (Primary) 14H40, 14K15 (Secondary) We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2 + b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem in this context and give a geometric proof of the fact that $P$ has quaternionic multiplication (QM) by the quaternion order of discriminant $6$. This allows us to describe the Galois action on the geometric endomorphism algebra of $P$. As an application, we classify the torsion subgroups of the Mordell-Weil groups $P(\mathbb{Q})$, as both abelian groups and $\text{End}(P)$-modules. |
| title | The geometry and arithmetic of bielliptic Picard curves |
| topic | Algebraic Geometry Number Theory 14H45 (Primary) 14H40, 14K15 (Secondary) |
| url | https://arxiv.org/abs/2308.15297 |