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

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 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 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 ⊗ f 2 ⊗ f 3 ⊗ 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.