Arithmetic level raising on triple product of Shimura curves and Gross--Kudla--Schoen Diagonal cycles II: Bipartite Euler system
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866911684490166272 |
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| author | Wang, Haining |
| author_facet | Wang, Haining |
| contents | In this article, we study the Gross--Kudla--Schoen diagonal cycle on the triple product of Shimura curves at a place of good reduction and prove an unramified arithmetic level raising theorem for the cohomology of this triple product. We deduce from it a reciprocity law which relates the image of the diagonal cycle under the Abel--Jacobi map to certain period integral of Gross--Kudla type. Combing this with the first reciprocity law we proved in a previous work, we show that the Gross--Kudla--Schoen diagonal cycles form a bipartite Euler system for the symmetric cube motive of a modular form. As an application we provide some evidence for the rank one case of the Bloch--Kato conjecture for the symmetric cube motive of a modular form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_14916 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Arithmetic level raising on triple product of Shimura curves and Gross--Kudla--Schoen Diagonal cycles II: Bipartite Euler system Wang, Haining Number Theory Algebraic Geometry In this article, we study the Gross--Kudla--Schoen diagonal cycle on the triple product of Shimura curves at a place of good reduction and prove an unramified arithmetic level raising theorem for the cohomology of this triple product. We deduce from it a reciprocity law which relates the image of the diagonal cycle under the Abel--Jacobi map to certain period integral of Gross--Kudla type. Combing this with the first reciprocity law we proved in a previous work, we show that the Gross--Kudla--Schoen diagonal cycles form a bipartite Euler system for the symmetric cube motive of a modular form. As an application we provide some evidence for the rank one case of the Bloch--Kato conjecture for the symmetric cube motive of a modular form. |
| title | Arithmetic level raising on triple product of Shimura curves and Gross--Kudla--Schoen Diagonal cycles II: Bipartite Euler system |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2004.14916 |