Twisted triple product root numbers and a cycle of Darmon-Rotger
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929735032897536 |
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| author | Lilienfeldt, David T. -B. G. |
| author_facet | Lilienfeldt, David T. -B. G. |
| contents | We consider an algebraic cycle on the triple product of the prime level modular curve $X_0(p)$ with origins in work of Darmon and Rotger. It is defined over the quadratic extension of $\mathbb{Q}$ ramified only at $p$ whose associated quadratic character $χ$ is the Legendre symbol at $p$. We prove that it is null-homologous and describe actions of various groups on it. For any three normalised cuspidal eigenforms $f_1, f_2, f_3$ of weight $2$ and level $Γ_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\otimes f_2\otimes f_3\otimes χ, s)$ is $-1$. Assuming conjectures of Beilinson and Bloch, and guided by the Gross-Zagier philosophy, this suggests that the Darmon-Rotger cycle could be non-torsion, although we do not currently have a proof of this. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twisted triple product root numbers and a cycle of Darmon-Rotger Lilienfeldt, David T. -B. G. Number Theory Algebraic Geometry 11G40, 11G18, 11F11, 14C25 We consider an algebraic cycle on the triple product of the prime level modular curve $X_0(p)$ with origins in work of Darmon and Rotger. It is defined over the quadratic extension of $\mathbb{Q}$ ramified only at $p$ whose associated quadratic character $χ$ is the Legendre symbol at $p$. We prove that it is null-homologous and describe actions of various groups on it. For any three normalised cuspidal eigenforms $f_1, f_2, f_3$ of weight $2$ and level $Γ_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\otimes f_2\otimes f_3\otimes χ, s)$ is $-1$. Assuming conjectures of Beilinson and Bloch, and guided by the Gross-Zagier philosophy, this suggests that the Darmon-Rotger cycle could be non-torsion, although we do not currently have a proof of this. |
| title | Twisted triple product root numbers and a cycle of Darmon-Rotger |
| topic | Number Theory Algebraic Geometry 11G40, 11G18, 11F11, 14C25 |
| url | https://arxiv.org/abs/2410.06063 |