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arXiv · 2501.14060

Ceresa Cycles of $X_{0}(N)$

Elvira Lupoian·James Rawson
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Abstract

The Ceresa cycle is an algebraic 1-cycle on the Jacobian of an algebraic curve. Although it is homologically trivial, Ceresa famously proved that for a very general complex curve of genus at least 3, it is non-trivial in the Chow group. In this paper we study the Ceresa cycle attached to the complete modular curve $X_{0}(N)$ modulo rational equivalence. For prime level $p$ we give a complete description, namely we prove that if $X_{0}(p)$ is not hyperelliptic, then its Ceresa cycle is non-torsion. For general level $N$, we prove that there are finitely many $X_{0}(N)$ with torsion Ceresa cycle. Our method relies on the relationship between the vanishing of the Ceresa cycle and Chow-Heegner points on the Jacobian. We use the geometry and arithmetic of modular Jacobians to prove that such points are of infinite order and therefore deduce non-vanishing of the Ceresa cycle.

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Elvira Lupoian, James Rawson. 2025-01-23. Ceresa Cycles of $X_{0}(N)$. https://arxiv.org/abs/2501.14060

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