On the torsion subgroups of the modular Jacobians

TL;DR

This paper investigates the structure of torsion subgroups of modular Jacobians, establishing bounds for their relation to cuspidal subgroups across various levels and characters.

Contribution

It provides explicit bounds relating the rational torsion subgroup to the cuspidal subgroup for all $J_0(N)$ and certain $J_0(DC)$, extending understanding of their structure.

Findings

Rational torsion subgroup of $J_0(N)$ matches its cuspidal subgroup up to a factor of $6N extstyleig( extstyle extstyleig)",

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Abstract

For any positive integer $N$, we prove that the rational torsion subgroup of $J_0(N)$ agrees with its rational cuspidal subgroups up to a factor of $6N\prod_{p\mid N}(p^2-1)$. Moreover, for modular Jacobians of the form $J_0(DC)$ with $D$ a positive square-free integer and $C$ any positive divisor of $D$, we prove that the $\psi$-part of the torsion subgroup of $J_0(DC)$ agrees with the $\psi$-part of its cuspidal subgroup up to a factor of $6D\prod_{p\mid D}(p^2-1)$, where $\psi$ is any quadratic character of conductor dividing $C$.