Ring of modular forms
In mathematics, a ring of modular forms is a graded ring generated by modular forms. Rings of modular forms give algebraic structure to the space of all modular forms of a given group.
Definition
Let Γ be a subgroup of SL(2, Z) that is of finite index and let Mk(Γ) be the ring of modular forms of weight k. The ring of modular forms of Γ is the graded ring .[1]
Properties
The ring of modular forms is a graded Lie algebra since the Lie bracket of modular forms f and g of respective weights k and ℓ is a modular form of weight k + ℓ + 2.[1] In fact, a bracket can be defined for the n-th derivative of modular forms and such a bracket is called a Rankin–Cohen brackets.[1]
![{\displaystyle [f,g]=kfg'-\ell f'g}](/media/api/rest_v1/media/math/render/svg/c772fadd99b976a03b990eeb3611f0768f42e3c0)
Pierre Deligne and Michael Rapaport showed that the ring of modular forms M(Γ) is finitely generated when Γ is a congruence subgroup of SL(2, Z).[2]
References
- ^ a b c Zagier, Don. "Elliptic Modular Forms and Their Applications" (PDF). In Bruinier, Jan Hendrik; van der Geer, Gerard; Harder, Günter; Zagier, Don (eds.). The 1-2-3 of Modular Forms. Universitext. Springer-Verlag. pp. 1–103. doi:10.1007/978-3-540-74119-0_1. ISBN 978-3-540-74119-0.
- ^ Deligne, Pierre; Rapaport, Michael (1973). "Les schémas de modules de courbes elliptiques". Modular functions of one variable, II. Lecture Notes in Mathematics. Vol. 349. Berlin: Springer-Verlag. pp. 143–316.
