Moduli interpretation of Eisenstein series

Details Moduli interpretation of Eisenstein series Moduli interpretation of Eisenstein series

2009-03-08

arxiv.org/abs/0903.1439v6

International Journal of Number Theory 8 (2012), no. 3, 715-748

Mathematics

Number Theory

29 pages, amslatex. Version 6: corrected a sign misprint in equation (4.6) (thanks to N. Mascot for pointing it out). Final ac

Scientific paper

10.1142/S1793042112500418

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.

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