Rational torsion points on Jacobians of modular curves

Hwajong Yoo¹

¹ Center for Geometry and Physics Institute for Basic Science (IBS) Pohang 37673, Republic of Korea

Acta Arithmetica, Tome 172 (2016) no. 4, pp. 299-304.

Résumé

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 $p\equiv 1(\mathrm{mod}9)$ and $3^{(p-1)/3}\equiv 1(\mathrm{mod}p)$.

Détail

DOI : 10.4064/aa8140-12-2015

Mots-clés : prime greater consider modular curve mathbb its jacobian variety mathbb mathcal mathcal group rational torsion points cuspidal group respectively prove primary subgroups mathcal mathcal coincide unless equiv pmod p equiv pmod