Ring of modular forms

In mathematics, the ring of modular forms associated to a subgroup Γ of the special linear group SL(2, Z) is a graded ring generated by the modular forms of Γ. The study of rings of modular forms describes the algebraic structure to 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 ring of modular forms of weight k. The ring of modular forms of Γ is the graded ring $M(\Gamma )=\bigoplus _{k>0}M_{k}(\Gamma )$ .

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 $\mathbb {C} [E_{4},E_{6}]$ , which is a polynomial ring of two variables.

Properties

The ring of modular forms is a graded Lie algebra since the Lie bracket $[f,g]=kfg'-\ell f'g$ of modular forms f and g of respective weights k and is a modular form of weight k + ℓ + 2. 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.

Congruence subgroups of SL(2, Z)

In 1973, 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).

In 2003, Lev Borisov and Paul Gunnells showed that the ring of modular forms M(Γ) is generated in weight at most 3 when $\Gamma$ is the congruence subgroup $\Gamma _{1}(N)$ 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 $\Gamma _{1}(N)$ to all levels N and also demonstrated that the ring of modular forms for the congruence subgroup $\Gamma _{0}(N)$ is generated in weight at most 6 for some levels N.

In 2016, Aaron Landesman, Peter Ruhm, and Robin Zhang further generalized these results and proved that the ring of modular forms of any congruence subgroup Γ of SL(2, Z) is generated in weight at most 6 with relations generated in weight at most 12, with even lower bounds of 5 and 10 when Γ has no nonzero odd weight modular forms.

General Fuchsian groups

A Fuchsian group Γ corresponds to the orbifold obtained from the quotient $\Gamma \backslash \mathbb {H}$ of the upper half-plane $\mathbb {H}$ . By a generalization of Serre's GAGA due to Jon Voight and David Zureick-Brown, there is a correspondence between the ring of modular forms of Γ and the a particular section ring closely related to the canonical ring of a stacky curve.

There is a general formula for the weights of generators and relations of rings of modular forms due to the work of Voight and Zureick-Brown and the work of Landesman, Ruhm, and Zhang. Let $e_{i}$ be the stabilizer orders of the stacky points of the stacky curve (equivalently, the cusps of the orbifold $\Gamma \backslash \mathbb {H}$ ) associated to Γ. If Γ has no nonzero odd weight modular forms, then the ring of modular forms is generated in weight at most $6\max(3,e_{1},e_{2},\ldots ,e_{r})$ and has relations generated in weight at most $12\max(3,e_{1},e_{2},\ldots ,e_{r})$ If Γ has a nonzero odd weight modular form, then the ring of modular forms is generated in weight at most $\max(5,e_{1},e_{2},\ldots ,e_{r})$ and has relations generated in weight at most $2\max(5,e_{1},e_{2},\ldots ,e_{r})$ .

Applications

In string theory and supersymmetric gauge theory, the algebraic structure of the ring of modular forms can be used to study the structure of the Higgs vacua of four-dimensional gauge theories with N = 1 supersymmetry. The stabilizers of superpotentials in N = 4 supersymmetric Yang–Mills theory are rings of modular forms of the congruence subgroup Γ(2) of SL(2, Z).