No abstract

No abstract

Suppose that φ(f, g) = 0. Then we have 0 = θ(Af + F (g)) = θ(Af ), hence 0 = θ(f ). But then f = 0, as the weight of f is not divisible by p. It follows that F (g) = 0, hence that g = 0. 4.5Now that we know how to characterize the image ofKatz , it becomes time to investigate how to compute this ambient vector space. We want to avoid the problems related to the lifting of elements of S p (Γ 1 (N), ε, F) ′ Katz to characteristic zero with a given character (see [22, §1] and [22, Prop. 1.10], they have to do with what is called Carayol's Lemma). So we describeKatz of characteristic zero forms of weight p with no prescribed character.Proposition 4.6 Let p be a prime number, and N ≥ 1 an integer not divisible by p. Suppose that N = 1 or that p ≥ 5.