2025/01/23 by Elvira Lupoian, Lupoian, Elvira, James Rawson +1 · 2 citations
Computer Science · Mathematics · #11G18 #14C25 #14G35 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Cellular Automata and Applications #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2501.14060
openalex publication_date 2025/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 X0(N) modulo rational equivalence. For prime level p we give a complete description, namely we prove that if X0(p) is not hyperelliptic, then its Ceresa cycle is non-torsion. For general level N, we prove that there are finitely many X0(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.