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, J(0)(N)(Q)(tor)[q(infinity)] = 0 if q is a prime not dividing 6 . N . Pi(p)(vertical bar N)(p(2) - 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. (C) 2018 Elsevier Inc. All rights reserved.