Another look at rational torsion of modular Jacobians

TL;DR

This paper investigates the structure of the rational torsion subgroup of modular Jacobians, providing a new proof of a key result relating it to the cuspidal subgroup using Hecke algebra modules.

Contribution

It offers a novel proof of Ohta's result on the rational torsion of $J_0(N)$, avoiding explicit cardinality computations by leveraging Hecke algebra module structure.

Findings

The $p$-primary part of the rational torsion equals that of the cuspidal subgroup for primes not dividing 6N.

The new proof simplifies understanding of torsion structure without explicit group order calculations.

The approach highlights the role of Hecke algebra modules in analyzing modular Jacobian torsion.

Abstract

We study the rational torsion subgroup of the modular Jacobian $J_0(N)$ for $N$ a square-free integer. We give a new proof of a result of Ohta on a generalization of Ogg's conjecture: for a prime number $p \nmid 6N$, the $p$-primary part of the rational torsion subgroup equals that of the cuspidal subgroup. Whereas previous proofs of this result used explicit computations of the cardinalities of these groups, we instead use their structure as modules for the Hecke algebra.