 
Summary: Rational torsion in elliptic curves and the cuspidal
subgroup
Amod Agashe
Abstract
Let A be an elliptic curve over Q of square free conductor N. Sup
pose A has a rational torsion point of prime order r such that r does
not divide 6N. We prove that then r divides the order of the cuspidal
subgroup C of J0(N). If A is optimal, then viewing A as an abelian
subvariety of J0(N), our proof shows more precisely that r divides
the order of A C. Also, under the hypotheses above, we show that
for some prime p that divides N, the eigenvalue of the AtkinLehner
involution Wp acting on the newform associated to A is 1.
1 Introduction
Let A be an elliptic curve over Q of square free conductor N and let A be
the optimal curve in the isogeny class (over Q) of A . Let X0(N) denote
the modular curve over Q associated to 0(N), and let J0(N) be its Ja
cobian. By [BCDT01], we may view A as an abelian variety quotient over
Q of J0(N). By dualizing, A can also be viewed as an abelian subvariety
of J0(N), as we shall do in the rest of this article. The cuspidal subgroup C
of J0(N)(C) is the group of degree zero divisors on X0(N)(C) that are sup
