{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/torsion-phenomena-for-zero-cycles-on-a","title":"Torsion phenomena for zero-cycles on a product of curves over a number field","arxiv_id":"2204.05876","date":"2022-04-12","proceeding":null,"authors":["Evangelia Gazaki","Jonathan Love"],"abstract":"For a smooth projective variety $X$ over a number field $k$ a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of $X$ is a torsion group. In this article we consider a product $X=C_1\\times\\cdots\\times C_d$ of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for $X$. Additionally, we produce many new examples of non-isogenous elliptic curves $E_1, E_2$ with positive rank over $\\mathbb{Q}$ for which the image of the natural map $E_1(\\mathbb{Q})\\otimes E_2(\\mathbb{Q})\\xrightarrow{\\varepsilon} \\text{CH}_0(E_1\\times E_2)$ is finite, including the first known examples of rank greater than $1$. Combining the two results, we obtain infinitely many nontrivial products $X=C_1\\times\\cdots\\times C_d$ for which the analogous map $\\varepsilon$ has finite image.","url_abs":"https://arxiv.org/abs/2204.05876v2","url_pdf":"https://arxiv.org/pdf/2204.05876v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"torsion-phenomena-for-zero-cycles-on-a","repo_url":"https://github.com/jonathanrlove/zero-cycles","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}