Generalized Jacobians of modular and Drinfeld modular curves

TL;DR

This paper explicitly determines the structure of rational torsion points on generalized Jacobians of modular and Drinfeld modular curves for square-free levels, revealing new differences from prime power levels.

Contribution

It provides the first explicit description of the rational torsion on generalized Jacobians for square-free levels and extends results to Drinfeld modular curves.

Findings

Rational torsion points are explicitly characterized for square-free levels.

The structure differs significantly from prime power level cases.

Results include an analysis of Hecke action and Eisenstein properties.

Abstract

We consider the generalized Jacobian $\widetilde{J}$ of the modular curve $X_0(N)$ of level $N$ with respect to a reduced divisor consisting of all cusps. Supposing $N$ is square free, we explicitly determine the structure of the $\mathbb{Q}$-rational torsion points on $\widetilde{J}$ up to $6$-primary torsion. The result turns out to be very different from the case of prime power level previously studied by Yang and the second author. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on $\widetilde{J}$ and its Eisenstein property.