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

TL;DR

This paper studies an algebraic cycle on the triple product of modular curves, showing it is null-homologous, and explores its implications for the root numbers of twisted triple product L-functions, suggesting potential non-torsion properties.

Contribution

It introduces a specific algebraic cycle related to Darmon and Rotger's work, proves its null-homologous nature, and analyzes the root number of associated L-functions under certain conjectures.

Findings

The cycle is null-homologous.

The root number of the twisted L-function is -1.

Potential non-torsion of the cycle under conjectural assumptions.

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.