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

We consider an algebraic cycle on the triple product of the prime level modular curve X₀(p) with origins in work of Darmon and Rotger. It is defined over the quadratic extension of 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 normalized cuspidal eigenforms f₁, f₂, f₃ of weight 2 and level Γ₀)(p), we prove that the global root number of the twisted triple product L-function L(f₁ ⊗ f₂ ⊗ f₃ ⊗ 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.