The Kernel of the Eisenstein Ideal Original Research Article

Let N be a prime number, and let J0(N) be the Jacobian of the modular curve X0(N). Let T denote the endomorphism ring of J0(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal Isubset of or equal toT, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J0(N)[I] of this ideal (the set of points in J0(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q/Q) on J0(N)[I]. Our results were previously known in the special case where N−1 is not divisible by 16.