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arXiv · 2410.06063

Twisted triple product root numbers and a cycle of Darmon-Rotger

David T. -B. G. Lilienfeldt
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Abstract

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 $\chi$ 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 $\Gamma_0(p)$, we prove that the global root number of the twisted triple product $L$-function $L(f_1\otimes f_2\otimes f_3\otimes \chi, s)$ 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.

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David T. -B. G. Lilienfeldt. 2024-10-08. Twisted triple product root numbers and a cycle of Darmon-Rotger. https://doi.org/10.1007/s11856-026-2932-5

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