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Abstract

We prove a conjecture of Coleman, Kaskel, and Ribet concerning the way in which the modular curve X₀(p) intersects the torsion subgroup of its Jacobian J₀(p) when the curve is embedded in its Jacobian by sending a point P to the class of (P)-(∞). Results are also obtained for certain composite levels, for noncuspidal embeddings of the modular curve in its Jacobian, and for certain quotients of X₀(N). Key tools include Ribet's level-lowering theorem, Mazur's work on the Eisenstein ideal, and results of Grothendieck on semistable abelian varieties.