Rational torsion subgroups of modular Jacobian varieties

In this article, we study the Q-rational torsion subgroups of the Jacobian varieties of modular curves. The main result is that, for any positive integer N, J0(N)(Q)tor[q∞]=0 if q is a prime not dividing 6⋅N⋅∏p|N(p2−1). To prove the result, we explicitly construct a collection of Eisenstein series with rational Fourier expansions, and then determine their constant terms to control the size of the rational torsion subgroups.