Ceresa cycles of bielliptic Picard curves
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929641291251712 |
|---|---|
| author | Laga, Jef Shnidman, Ari |
| author_facet | Laga, Jef Shnidman, Ari |
| contents | We show that the Ceresa cycle $κ(C_t)$ of the genus $3$ curve $C_t \colon y^3 = x^4 + 2tx^2 + 1$ is torsion if and only if $Q_t=( \sqrt[3]{t^2 -1},t)$ is a torsion point on the elliptic curve $y^2 = x^3 + 1$. This shows that there are infinitely many smooth plane quartic curves over $\mathbb{C}$ (resp. $\mathbb{Q}$) with torsion (resp. infinite order) Ceresa cycle. Over $\overline{\mathbb{Q}}$, we show that the Beilinson--Bloch height of $κ(C_t)$ is proportional to the Neron--Tate height of $Q_t$. Thus, the height of $κ(C_t)$ is nondegenerate and satisfies a Northcott property. To prove all this, we show that the Chow motive that controls $κ(C_t)$ is isomorphic to $\mathfrak{h}^1$ of an appropriate elliptic curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12965 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ceresa cycles of bielliptic Picard curves Laga, Jef Shnidman, Ari Algebraic Geometry Number Theory 14C25 (Primary) 14H40, 14H45 (Secondary) We show that the Ceresa cycle $κ(C_t)$ of the genus $3$ curve $C_t \colon y^3 = x^4 + 2tx^2 + 1$ is torsion if and only if $Q_t=( \sqrt[3]{t^2 -1},t)$ is a torsion point on the elliptic curve $y^2 = x^3 + 1$. This shows that there are infinitely many smooth plane quartic curves over $\mathbb{C}$ (resp. $\mathbb{Q}$) with torsion (resp. infinite order) Ceresa cycle. Over $\overline{\mathbb{Q}}$, we show that the Beilinson--Bloch height of $κ(C_t)$ is proportional to the Neron--Tate height of $Q_t$. Thus, the height of $κ(C_t)$ is nondegenerate and satisfies a Northcott property. To prove all this, we show that the Chow motive that controls $κ(C_t)$ is isomorphic to $\mathfrak{h}^1$ of an appropriate elliptic curve. |
| title | Ceresa cycles of bielliptic Picard curves |
| topic | Algebraic Geometry Number Theory 14C25 (Primary) 14H40, 14H45 (Secondary) |
| url | https://arxiv.org/abs/2312.12965 |