Rational torsion points on Jacobians of modular curves
Volume 172 / 2016
Acta Arithmetica 172 (2016), 299-304
MSC: 11G10, 11G18, 14G05.
DOI: 10.4064/aa8140-12-2015
Published online: 3 December 2015
Abstract
Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb Q$ and its Jacobian variety $J_0(3p)$ over $\mathbb Q$. Let $\mathcal T(3p)$ and $\mathcal C(3p)$ be the group of rational torsion points on $J_0(3p)$ and the cuspidal group of $J_0(3p)$, respectively. We prove that the $3$-primary subgroups of $\mathcal T(3p)$ and $\mathcal C(3p)$ coincide unless $p\equiv 1 \pmod 9$ and $3^{(p-1)/3} \equiv 1 \pmod {p}$.
