In mathematics, the ring of modular forms associated to a subgroup Γ of the special linear group SL(2, Z) is the graded ring generated by the modular forms of Γ. The study of rings of modular forms describes the algebraic structure of the space of modular forms.
Definition Let Γ be a subgroup of SL(2, Z) that is of finite index and let Mk(Γ) be the vector space of modular forms of weight k. The ring of modular forms of Γ is the graded ring M ( Γ ) = ⨁ k ≥ 0 M k ( Γ ) {\textstyle M(\Gamma )=\bigoplus _{k\geq 0}M_{k}(\Gamma )} .
Example The ring of modular forms of the full modular group SL(2, Z) is freely generated by the Eisenstein series E4 and E6. In other words, Mk(Γ) is isomorphic as a C {\displaystyle \mathbb {C} } -algebra to C [ E 4 , E 6 ] {\displaystyle \mathbb {C} [E_{4},E_{6}]} , which is the polynomial ring of two variables over the complex numbers.
Properties The ring of modular forms is a graded Lie algebra since the Lie bracket [ f , g ] = k f g ′ − ℓ f ′ g {\displaystyle [f,g]=kfg'-\ell f'g} of modular forms f and g of respective weights k and ℓ is a modular form of weight k + ℓ + 2. A bracket can be defined for the n-th derivative of modular forms and such a bracket is called a Rankin–Cohen bracket.
Congruence subgroups of SL(2, Z) In 1973, Pierre Deligne and Michael Rapoport showed that the ring of modular forms M(Γ) is finitely generated when Γ is a congruence subgroup of SL(2, Z). In 2003, Lev Borisov and Paul Gunnells showed that the ring of modular forms M(Γ) is generated in weight at most 3 when Γ {\displaystyle \Gamma } is the congruence subgroup Γ 1 ( N ) {\displaystyle \Gamma _{1}(N)} of prime level N in SL(2, Z) using the theory of toric modular forms. In 2014, Nadim Rustom extended the result of Borisov and Gunnells for Γ 1 ( N ) {\displaystyle \Gamma _{1}(N)} to all levels N and also demonstrated that the ring of modular forms for the congruence subgroup Γ 0 ( N ) {\displaystyle \Gamma _{0}(N)} is generated in weight at most 6 for some levels N. In 2015, John Voight and David Zureick-Brown generalized these results: they proved that the graded ring of modular forms of even weight for any congruence subgroup Γ of SL(2, Z) is generated in weight at most 6 with relations generated in weight at most 12. Building on this work, in 2016, Aaron Landesman, Peter Ruhm, and Robin Zhang showed that the same bounds hold for the full ring (all weights), with the improved bounds of 5 and 10 when Γ has some nonzero odd weight modular form.
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