Rational torsion points on Jacobians of modular curves

TL;DR

This paper investigates the structure of rational torsion points on Jacobians of modular curves, establishing conditions under which the 3-primary parts of torsion and cuspidal groups coincide, revealing new insights into their arithmetic properties.

Contribution

It proves that the 3-primary subgroups of the rational torsion and cuspidal groups on J_0(3p) are equal except for specific congruence conditions on p.

Findings

The 3-primary parts of torsion and cuspidal groups coincide unless p ≡ 1 mod 9 and 3^{(p-1)/3} ≡ 1 mod p.

Identifies explicit conditions where the torsion and cuspidal groups differ.

Enhances understanding of the torsion subgroup structure of Jacobians of modular curves.

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