Faber--Pandharipande cycle, real multiplication and torsion points

TL;DR

This paper proves a conjecture relating Faber--Pandharipande cycles to torsion points on Shimura curves with real multiplication, connecting algebraic cycles, number fields, and special points on curves.

Contribution

It establishes the vanishing of Faber--Pandharipande cycles over number fields for Shimura curves with real multiplication, confirming a conjecture of Beilinson and Bloch in this context.

Findings

Faber--Pandharipande cycle vanishes over number fields for Shimura curves with real multiplication.

Connection established between these cycles and torsion points via the Abel--Jacobi map.

Method extends to other curves with partial real multiplication.

Abstract

A result of Green and Griffiths states that for the generic curve $C$ over $\mathbb{C}$ of genus $g \geq 4$ with a canonical divisor $K$, its Faber--Pandharipande 0-cycle $K\times K-(2g-2)K_\Delta$ 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.