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arXiv · 2407.20998

Non-vanishing of Ceresa and Gross--Kudla--Schoen cycles associated to modular curves

Matt Kerr·Wanlin Li·Congling Qiu·Tonghai Yang
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Abstract

Associated to an algebraic curve $X$, there are two canonically constructed homologically trivial algebraic $1$-cycles, the Ceresa cycle in the Jacobian of $X$, and the Gross-Kudla-Schoen modified diagonal cycle in the triple product $X \times X \times X$. By a result of Shou-Wu Zhang, one is torsion if and only if the other is. In this paper, we prove that these two cycles associated to a large family of modular curves are non-torsion in the corresponding Chow groups. We obtain the result by relating this problem to the study of special cycles on orthogonal Shimura varieties. As the main ingredient and a result of independent interest, we develop a pullback formula for special divisors on modular curves embedded in their products via the diagonal map.

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Matt Kerr, Wanlin Li, Congling Qiu, Tonghai Yang. 2024-07-30. Non-vanishing of Ceresa and Gross--Kudla--Schoen cycles associated to modular curves. https://arxiv.org/abs/2407.20998

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