MODULI INTERPRETATION OF EISENSTEIN SERIES

Abstract

Let ℓ ≥ 3. Using the moduli interpretation, we define certain elliptic modular forms of level Γ(ℓ) over any field k where 6ℓ is invertible and k contains the ℓth roots of unity. These forms generate a graded algebra [Formula: see text], which, over C, is generated by the Eisenstein series of weight 1 on Γ(ℓ). The main result of this article is that, when k = C, the ring [Formula: see text] contains all modular forms on Γ(ℓ) in weights ≥ 2. The proof combines algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions. Our results give a systematic 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.