Moduli Interpretation of Eisenstein Series

Let l ≥ 3. Using the moduli interpretation, we define certain elliptic modular forms of level Γ(l), which make sense over any field k in which 6l 6= 0 and that contains the lth roots of unity. Over the complex numbers, these forms include all holomorphic Eisenstein series on Γ(l) in all weights, in a natural way. The graded ring Rl that is generated by our special modular forms turns out to be generated by certain forms in weight 1 that, over C, correspond to the Eisenstein series on Γ(l). By a combination of algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions, we show that when k = C, the ring Rl, which is generated as a ring by the Eisenstein series of weight 1, contains all modular forms on Γ(l) in weights ≥ 2. Our results give a straightforward method to produce models for the modular curve X(l) defined over the lth cyclotomic field, using only exact arithmetic in the l-torsion field of a single Q-rational elliptic curve E0.