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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.08989 |
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Table of Contents:
- 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 $K\times K-(2g-2)K_Δ$ 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.