  Let L >= 3. Using the moduli interpretation, we define certain elliptic
modular forms of level Gamma(L) over any field k where 6L is invertible and k
contains the Lth roots of unity. These forms generate a graded algebra R_L,
which, over C, is generated by the Eisenstein series of weight 1 on Gamma(L).
The main result of this article is that, when k=C, the ring R_L contains all
modular forms on Gamma(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 E^0.
