Faber--Pandharipande cycle, real multiplication and torsion points

A result of Green and Griffiths states that for the generic curve $C$ over $\mathbb{C}$ of genus $g\ge 4$ with a canonical divisor $K$, its Faber–Pandharipande 0-cycle $K\times K-(2g-2)K_{\mathrm{\Delta }}$ on $C\times C$ is nontorsion in the Chow group of rational equivalence classes. However, according to a conjecture of Beilinson and Bloch, this Chow cycle vanishes if the curve is defined over a number field. We give a proof of this prediction for Shimura curves which have real multiplication. Our method also works for some other classes curves with partial real multiplication. We also draw a connection between the Faber–Pandharipande 0-cycles and torsion points on curves under the Abel–Jacobi map.