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Rational torsion points on Jacobians of modular curves

Volume 172 / 2016

Hwajong Yoo 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 $3^{(p-1)/3} \equiv 1 \pmod {p}$.

Authors

  • Hwajong YooCenter for Geometry and Physics
    Institute for Basic Science (IBS)
    Pohang 37673, Republic of Korea
    e-mail

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