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Amod Agashe
Rational torsion in elliptic curves and the cuspidal subgroup, to appear, online on 13 December 2017, 11 p.
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Abstract

Let $A$ be an elliptic curve over $\mathbb{Q}$ of square free conductor $N$ that has a rational torsion point of prime order $r$ such that $r$ does not divide $6N$. We show that then $r$ divides the order of the cuspidal subgroup $C$ of $J_0(N)$. If $A$ is optimal, then viewing $A$ as an abelian subvariety of $J_0(N)$, our proof shows more precisely that $r$ divides the order of $A \cap C$. Also, under the hypotheses above minus the hypothesis that $r$ does not divide $N$, we show that for some prime $p$ that divides $N$, the eigenvalue of the Atkin–Lehner involution $W_p$ acting on the newform associated to $A$ is $-1$.

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