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

Elvira Lupoian, James Rawson

TL;DR

The paper analyzes the Ceresa cycle on Jacobians of modular curves, focusing on $X_0(N)$. It establishes that for prime level $p$, the Ceresa cycle $ ext{Cer}(p)$ is non-torsion precisely when $X_0(p)$ is non-hyperelliptic, yielding concrete non-vanishing results for many primes; it also proves that only finitely many levels $N$ yield a torsion Ceresa cycle. The core method links vanishing of the Ceresa cycle to Chow-Heegner points (via GK-S cycles and shadow points) and leverages the endomorphism action of the modular Jacobians (notably Hecke operators) to produce infinite-order points. The results connect to triple product $L$-functions through height pairings and offer explicit computations (and computational criteria) for many levels, while detailing limitations when $J_0(N)(Q)$ has rank zero. Overall, the work combines modular-curve geometry, CM points, and endomorphism-technology to advance understanding of algebraic nontriviality of Ceresa cycles in a central arithmetic setting.

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.

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

TL;DR

The paper analyzes the Ceresa cycle on Jacobians of modular curves, focusing on . It establishes that for prime level , the Ceresa cycle is non-torsion precisely when is non-hyperelliptic, yielding concrete non-vanishing results for many primes; it also proves that only finitely many levels yield a torsion Ceresa cycle. The core method links vanishing of the Ceresa cycle to Chow-Heegner points (via GK-S cycles and shadow points) and leverages the endomorphism action of the modular Jacobians (notably Hecke operators) to produce infinite-order points. The results connect to triple product -functions through height pairings and offer explicit computations (and computational criteria) for many levels, while detailing limitations when has rank zero. Overall, the work combines modular-curve geometry, CM points, and endomorphism-technology to advance understanding of algebraic nontriviality of Ceresa cycles in a central arithmetic setting.

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 modulo rational equivalence. For prime level we give a complete description, namely we prove that if is not hyperelliptic, then its Ceresa cycle is non-torsion. For general level , we prove that there are finitely many 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.
Paper Structure (20 sections, 33 theorems, 18 equations, 3 tables)

This paper contains 20 sections, 33 theorems, 18 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Gross-Kudla-Schoen Cycles and Chow-Heegner Points
  3. The Ceresa Cycle and Gross-Kudla-Schoen Classes
  4. Chow-Heegner and Shadow Points
  5. Action of the Endomorphism Algebra
  6. The Arithmetic and Geometry of Modular Curves
  7. Torsion Subgroups of Modular Jacobians
  8. Canonical Divisors
  9. The Minimal Regular Models of $X_{0}(p)$ and $X_{0}^{+}(p)$
  10. Fixed Points of Hecke Operators
  11. Proof of Theorem 1
  12. Reduction Arguments
  13. Primes congruent to 5 or 7 modulo 12
  14. Primes congruent to 1 modulo 12
  15. Primes congruent to 11 modulo 12
  16. ...and 5 more sections

Key Result

Theorem 1

The Ceresa cycle $\mathop{\mathrm{Cer}}\nolimits(p)$ is non-zero if and only if $X_0(p)$ is not hyperelliptic. That is, if $p > 71$ or $p \in \{43, 53, 61, 67\}$, then $\mathop{\mathrm{Cer}}\nolimits(p)$ is non-trivial.

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Theorem 3: Zhang
  • Definition 2
  • Proposition 4
  • proof
  • Definition 3
  • Proposition 5
  • proof
  • ...and 46 more