Moduli interpretation of Eisenstein series

Khuri-Makdisi, Kamal
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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.
Comment: 29 pages, amslatex. Version 6: corrected a sign misprint in equation (4.6) (thanks to N. Mascot for pointing it out). Final accepted version
Mathematics - Number Theory, Mathematics - Algebraic Geometry, 11F11, 14H52, 14K10, 11F67, 11F25, 11G18