# The rational cuspidal divisor class group of $X_0(N)$

## TL;DR

This paper determines the structure of the rational cuspidal divisor class group of modular curves $X_0(N)$, constructing explicit divisors and computing their orders to understand the group's composition.

## Contribution

It explicitly constructs rational cuspidal divisors for each prime dividing $N$ and describes the entire structure of the rational cuspidal divisor class group.

## Key findings

- The $ ext{l}$-primary subgroup is a direct sum of cyclic groups.
- Constructed divisors generate the entire $ ext{l}$-primary subgroup.
- Confirmed conjectural equality with the rational torsion subgroup of $J_0(N)$.

## Abstract

For any positive integer $N$, we completely determine the structure of the rational cuspidal divisor class group of $X_0(N)$, which is conjecturally equal to the rational torsion subgroup of $J_0(N)$. More specifically, for a given prime $\ell$, we construct a rational cuspidal divisor $Z_\ell(d)$ for any non-trivial divisor $d$ of $N$. Also, we compute the order of the linear equivalence class of the divisor $Z_\ell(d)$ and show that the $\ell$-primary subgroup of the rational cuspidal divisor class group of $X_0(N)$ is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of the divisors $Z_\ell(d)$.