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

TL;DR

This paper proves that certain modified diagonal cycles on triple products of modular curves have torsion Abel-Jacobi images under specific conditions, revealing new torsion properties in algebraic cycles and Chow-Heegner points.

Contribution

It establishes torsion results for modified diagonal cycles on triple products of modular curves, extending to both complex and étale Abel-Jacobi maps, with applications to Chow-Heegner points.

Findings

Torsion property of Abel-Jacobi images of modified diagonal cycles.

Results hold for prime and square-free levels.

Applications to torsion in Chow-Heegner points on elliptic curves.

Abstract

Consider three normalised 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 \'etale counterpart. As an application, we deduce torsion properties of Chow-Heegner points associated with modified diagonal cycles on elliptic curves of prime level with split multiplicative reduction. The approach also works in the case of composite square-free level.