On the ramification of Hecke algebras at Eisenstein primes

Abstract

Fix a prime p, and a modular residual representation ρ : GQ → GL2(Fp). Suppose f is a normalized cuspidal Hecke eigenform of some level N and weight k that gives rise to ρ, and let Kf denote the extension of Qp generated by the q-expansion coefficients an(f) of f . The field Kf is a finite extension of Qp. What can one say about the extension Kf/Qp? Buzzard [1] has made the following conjecture: if N is fixed, and k is allowed to vary, then the degree [Kf : Qp] is bounded independently of k. Little progress has been made on this conjecture so far; indeed, very little seems to have been proven at all regarding the degrees [Kf : Qp]. The goal of this paper is to consider a question somewhat orthogonal to that of Buzzard, namely, to fix the weight and vary the level. Moreover, we only consider certain reducible representations ρ that arise in Mazur’s study of the Eisenstein Ideal [7]. Our results suggest that the degrees [Kf : Qp] are, in fact, arithmetically significant. Suppose that N ≥ 5 is prime, and that p is a prime which exactly divides the numerator of (N−1)/12. Mazur ([7], Prop. 9.6, p. 96 and Prop. 19.1, p. 140) has shown that there is a weight two normalized cuspidal Hecke eigenform defined over Qp, unique up to conjugation by GQp (the Galois group of Qp over Qp), satisfying the congruence a`(f) ≡ 1 + ` mod p (1)