Moduli interpretation of Eisenstein series

TL;DR

This paper uses moduli interpretation to construct and analyze Eisenstein series and modular forms on Gamma(L), providing a systematic approach to model modular curves over cyclotomic fields with explicit arithmetic methods.

Contribution

It introduces a new algebraic framework for generating modular forms of level Gamma(L) over arbitrary fields, extending classical results over complex numbers.

Findings

The algebra R_L is generated by Eisenstein series of weight 1 over any suitable field.

Over C, R_L contains all modular forms of weights >= 2 on Gamma(L).

The method allows explicit models of modular curves over cyclotomic fields.

Abstract

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.