Torsion properties of modified diagonal classes on triple products of modular curves

Abstract

AbstractConsider three normalized cuspidal eigenforms of weight $2$ and prime level p. Under the assumption that the global root number of the associated triple product L-function is $+1$ , we prove that the complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.