Rational torsion points on Jacobians of modular curves

Hwajong Yoo

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arXiv · 1504.04842

Rational torsion points on Jacobians of modular curves

Hwajong Yoo

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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 $3^{\frac{p-1}{3}} \equiv 1 \!\pmod {p}$.

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Hwajong Yoo. 2015-04-19. Rational torsion points on Jacobians of modular curves. https://arxiv.org/abs/1504.04842

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