Rational torsion of generalised modular Jacobians of odd level

TL;DR

This paper determines the structure of the rational torsion subgroup of generalized modular Jacobians of odd level, extending previous results to more general levels and providing detailed torsion subgroup decompositions.

Contribution

It explicitly describes the rational torsion subgroup of generalized modular Jacobians for odd levels, including prime-power and squarefree cases, up to 2-primary and l-primary parts.

Findings

Determined the torsion subgroup structure for odd levels.

Extended previous results to more general levels.

Provided explicit decompositions of torsion parts.

Abstract

We consider the generalised Jacobian $J_{0}(N)_{\mathbf{m}}$ of the modular curve $X_{0}(N)$ of level $N$, with respect to the modulus $\mathbf{m}$ consisting of all cusps on the modular curve. When $N$ is odd, we determine the group structure of the rational torsion $J_{0}(N)_{\mathbf{m}}(\mathbb{Q})_{tor}$ up to $2$-primary and $l$-primary parts for any prime $l$ dividing $N$. Our results extend those of Wei--Yamazaki for squarefree levels and Yamazaki--Yang for prime-power levels.