Quadratic torsion subgroups of modular Jacobian varieties

TL;DR

This paper investigates the structure of quadratic torsion subgroups of modular Jacobian varieties, showing that under certain conditions, these subgroups align with their cuspidal parts, except at specific primes.

Contribution

It establishes a precise correspondence between quadratic torsion subgroups and cuspidal subgroups for modular Jacobians, clarifying their structure away from problematic primes.

Findings

Quadratic torsion subgroups match cuspidal subgroups away from bad primes.

The result applies to Jacobians of modular curves with square-free levels.

The paper identifies primes where congruences affect the torsion structure.

Abstract

Let $D$ be an odd square-free positive integer and $C$ a divisor of $D$. For any quadratic character $\chi$ modulo $C$, we prove that the $\chi$-part of the group $J_0(DC)_\text{tor}$ of torsion points of $J_0(DC)$ coincides with the $\chi$-part of its cuspidal subgroup, away from those primes of bad reduction or where possible congruences between oldforms and newforms occur.