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

TL;DR

This paper investigates the Ceresa cycle on modular curves $X_{0}(N)$, demonstrating non-torsion properties for prime levels and finitely many torsion cases for general levels, using modular Jacobian arithmetic.

Contribution

It provides a complete description of the Ceresa cycle for prime levels and establishes finiteness results for torsion cases at general levels, linking geometry and arithmetic.

Findings

Ceresa cycle is non-torsion for non-hyperelliptic prime levels.

Finitely many $X_{0}(N)$ have torsion Ceresa cycle.

Uses Chow-Heegner points and modular Jacobian properties.

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.