Moduli of Elliptic Curves

The purpose of these notes is to provide a quick introduction to the moduli of elliptic curves. There are many excellent and thorough references on the subject, ranging from the slightly archaic [Igu59] and [Shi94] to the more difficult [KM85] and [DR73]. Brian’s forthcoming book on the Ramanujan conjecture also covers some of this material and includes a careful comparison of the transcendental and algebraic theories. In order to read Mazur’s paper [Maz78], it is not necessary to consider many of the subtle issues of the subject; indeed, all we will need to understand for the time being is how to construct good arithmetic models of Y1(N)/C, X1(N)/C, and the associated modular correspondences over Z[1/N ]. In order to have a concrete example or two to keep in mind, let us start by recalling the Tate normal form that was introduced in Bryden’s lecture. He discussed it in the following context: let K be a field and let E/K be an elliptic curve. Fix a point P ∈ E(K) such that P does not have order 1, 2, or 3. Bryden explained that there is a unique isomorphism of the pair (E/K,P ) to a pair (E(b, c)/K, (0, 0)), where b and c are in K and E(b, c) is given by the Weierstrass equation E(b, c) : y + (1− c)xy − by = x − bx.