The Kernel of the Eisenstein Ideal

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 I⊆T, 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.