Optimal quotients of modular Jacobians

Abstract

Let N be a positive integer, and f a normalized newform of weight two on Γ0(N). Attached to f is an optimal quotient Af of the Jacobian J0(N). We prove two theorems concerning such optimal quotients. (A) Let 𝕋 denote the algebra of endomorphisms of J0(N) generated by the Hecke operators, and let 𝔉f denote the ideal of fusion of f in 𝕋. If Âf denotes the dual abelian variety to Af, then the canonical polarization of J0(N) induces a polarization θf: Âf→Af. We show that there is an embedding ker(θf)⊂J0(N) [𝔉f] whose cokernel is supported at maximal ideals 𝔪 of 𝕋 for which J0(N)[𝔪] is not two-dimensional. (B) If N is prime, let C denote the subgroup of J0(N) generated by the divisor 0−∞ on X0(N). Mazur has shown that C is equal to the full torsion subgroup of J0(N)(ℚ), and that specialization modulo N induces an isomorphism of C with the group of connected components ΦJ0(N) of the characteristic N fibre of the Neron model of J0(N). We prove that analogous results hold for every optimal quotient of prime conductor, thereby generalizing results of Mestre and Oesterle (who treated the case of strong Weil curves) and confirming William Stein’s refined Eisenstein conjecture. The key idea in the proof of these two theorems is encapsulated in corollary 2.5 below, which allows us to apply multiplicity one results in a novel way to the study of optimal quotients.