On the Modularity of Elliptic Curves over Q: Wild 3-adic Exercises

The quotient of the upper half plane by Γ1(N), acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve Y1(N)/C. This curve has a natural model Y1(N)/Q, which for N > 3 is a fine moduli scheme for elliptic curves with a point of exact order N . We will let X1(N) denote the smooth projective curve which contains Y1(N) as a dense Zariski open subset. Recall that a cusp form of weight k ≥ 1 and level N ≥ 1 is a holomorphic function f on the upper half complex plane H such that • for all matrices ( a b c d ) ∈ Γ1(N)