Non-vanishing of Ceresa and Gross--Kudla--Schoen cycles associated to modular curves
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916798164631552 |
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| author | Kerr, Matt Li, Wanlin Qiu, Congling Yang, Tonghai |
| author_facet | Kerr, Matt Li, Wanlin Qiu, Congling Yang, Tonghai |
| contents | Associated to an algebraic curve $X$, there are two canonically constructed homologically trivial algebraic $1$-cycles, the Ceresa cycle in the Jacobian of $X$, and the Gross-Kudla-Schoen modified diagonal cycle in the triple product $X \times X \times X$. By a result of Shou-Wu Zhang, one is torsion if and only if the other is. In this paper, we prove that these two cycles associated to a large family of modular curves are non-torsion in the corresponding Chow groups. We obtain the result by relating this problem to the study of special cycles on orthogonal Shimura varieties. As the main ingredient and a result of independent interest, we develop a pullback formula for special divisors on modular curves embedded in their products via the diagonal map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-vanishing of Ceresa and Gross--Kudla--Schoen cycles associated to modular curves Kerr, Matt Li, Wanlin Qiu, Congling Yang, Tonghai Algebraic Geometry Number Theory Primary 14C25, Secondary 14G35 Associated to an algebraic curve $X$, there are two canonically constructed homologically trivial algebraic $1$-cycles, the Ceresa cycle in the Jacobian of $X$, and the Gross-Kudla-Schoen modified diagonal cycle in the triple product $X \times X \times X$. By a result of Shou-Wu Zhang, one is torsion if and only if the other is. In this paper, we prove that these two cycles associated to a large family of modular curves are non-torsion in the corresponding Chow groups. We obtain the result by relating this problem to the study of special cycles on orthogonal Shimura varieties. As the main ingredient and a result of independent interest, we develop a pullback formula for special divisors on modular curves embedded in their products via the diagonal map. |
| title | Non-vanishing of Ceresa and Gross--Kudla--Schoen cycles associated to modular curves |
| topic | Algebraic Geometry Number Theory Primary 14C25, Secondary 14G35 |
| url | https://arxiv.org/abs/2407.20998 |