Reprints available upon request.

Abstract

We give a new proof of Kamienny's Criterion, which is used in Merel's proof of the uniform boundedness theorem for rational torsion points on elliptic curves. This is done by relating the linear independence of certain Hecke operators to the fact that certain p-adic integrals on a symmetric power of X₀(N) have no common zeros. The main motivation for this paper was to demonstrate how the method of Coleman and Chabauty can be used to study rational points on symmetric powers of curves.