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The rational cuspidal subgroup of J_0(N)

Hwajong Yoo, Myungjun Yu

TL;DR

This work determines the rational cuspidal subgroup of J_0(N) in cases where the largest square dividing N, denoted ${L}$, is either an odd prime power or a product of two odd prime powers, showing ${\mathscr{C}}_N({\mathbf{Q}})={\mathscr{C}}(N)$. The authors develop a comprehensive framework of modular units on X_0(N) built from $F_{m,h}$ and $E_{g,h}$, and introduce a sophisticated map $\Psi$ that eliminates redundant factors to reduce to the NEW part. Central to the argument are vanishing results that force integrality of exponent vectors $E(m,h)$, enabling a precise control of the rational cuspidal subgroup and proving the claimed equality with the full cuspidal divisor class group in the stated cases. The results extend the program initiated by Yoo (and GYYY) to explicitly determine ${\mathscr{C}}_N({\mathbf{Q}})$ in non-squarefree settings, with implications for the structure of rational torsion in $J_0(N)$ and the broader landscape of modular-curves arithmetic.

Abstract

For a positive integer $N$, let $J_0(N)$ be the Jacobian of the modular curve $X_0(N)$. In this paper we completely determine the structure of the rational cuspidal subgroup of $J_0(N)$ when the largest perfect square dividing $N$ is either an odd prime power or a product of two odd prime powers. Indeed, we prove that the rational cuspidal divisor class group of $X_0(N)$ is the whole rational cuspidal subgroup of $J_0(N)$ for such an $N$, and the structure of the former group is already determined by the first author in [14].

The rational cuspidal subgroup of J_0(N)

TL;DR

This work determines the rational cuspidal subgroup of J_0(N) in cases where the largest square dividing N, denoted , is either an odd prime power or a product of two odd prime powers, showing . The authors develop a comprehensive framework of modular units on X_0(N) built from and , and introduce a sophisticated map that eliminates redundant factors to reduce to the NEW part. Central to the argument are vanishing results that force integrality of exponent vectors , enabling a precise control of the rational cuspidal subgroup and proving the claimed equality with the full cuspidal divisor class group in the stated cases. The results extend the program initiated by Yoo (and GYYY) to explicitly determine in non-squarefree settings, with implications for the structure of rational torsion in and the broader landscape of modular-curves arithmetic.

Abstract

For a positive integer , let be the Jacobian of the modular curve . In this paper we completely determine the structure of the rational cuspidal subgroup of when the largest perfect square dividing is either an odd prime power or a product of two odd prime powers. Indeed, we prove that the rational cuspidal divisor class group of is the whole rational cuspidal subgroup of for such an , and the structure of the former group is already determined by the first author in [14].

Paper Structure

This paper contains 15 sections, 50 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Modular units
  3. Motivational examples
  4. The map $\Psi$
  5. The definition of $\Psi_{\mathfrak{m}}$
  6. The definition of $\Psi$
  7. Formulas for $\Psi_{\mathfrak{m}}$
  8. Vanishing result I
  9. Vanishing result II
  10. Proof of Theorem \ref{['theorem: vanishing i=1']}
  11. Proof of Theorem \ref{['theorem: vanishing i=2']}
  12. Proof of Theorem \ref{['theorem A']}
  13. Case 1: $i=2$
  14. Case 2: $i=1$
  15. Case 3: $i=0$

Key Result

Theorem 1.3

Let $N=ML^2$ for some squarefree integer $M$. If $L$ is either an odd prime power or a product of two odd prime powers, then we have

Theorems & Definitions (107)

  • Conjecture 1.1: Generalized Ogg's conjecture
  • Conjecture 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • proof
  • Remark 1.9
  • Conjecture 1.10
  • ...and 97 more