Rational torsion in elliptic curves and the cuspidal subgroup

TL;DR

This paper proves a relationship between rational torsion points of prime order on elliptic curves over rationals and the cuspidal subgroup of modular Jacobians, under specific divisibility conditions.

Contribution

It establishes a new link between rational torsion points of prime order and the structure of the cuspidal subgroup for elliptic curves with square-free conductor.

Findings

Prime order torsion points imply divisibility of cuspidal subgroup order

Conditions on prime order and conductor are crucial

Enhances understanding of torsion structures in elliptic curves

Abstract

Let $A$ be an elliptic curve over $\Q$ of square free conductor $N$. We prove that if $A$ has a rational torsion point of prime order $r$ such that $r$ does not divide $6N$, then $r$ divides the order of the cuspidal subgroup of $J_0(N)$.