Torsion Subgroups of Modular Jacobian Varieties Over Splitting Fields of Cuspidal Subgroups

Abstract

Abstract For any positive integer $N$, let $J_0(N)$ be the Jacobian variety of the modular curve $X_0(N)$ over ${\mathbb {Q}}$ and ${\mathcal {C}}_N$ its cuspidal subgroup. Let $F_N$ denote the splitting field of ${\mathcal {C}}_N$, which is the smallest number field whose absolute Galois group acts trivially on ${\mathcal {C}}_N$. Let ${\mathcal {J}}_N=J_0(N)(F_{N})_{\textrm {tor}}$ be the torsion subgroup of the group of $F_N$-rational points on $J_0(N)$. We prove that ${\mathcal {J}}_N$ coincides with ${\mathcal {C}}_N$ outside $6N[F_N:{\mathbb {Q}}]$.