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arXiv · 2409.08989

Faber--Pandharipande cycle, real multiplication and torsion points

Congling Qiu
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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.

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Congling Qiu. 2024-09-13. Faber--Pandharipande cycle, real multiplication and torsion points. https://arxiv.org/abs/2409.08989

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