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 of the space of modular forms.
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 -algebra , which is a polynomial ring of two variables.
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. 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)Edit
In 2003, Lev Borisov and Paul Gunnells showed that the ring of modular forms M(Γ) is generated in weight at most 3 when is the congruence subgroup 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 to all levels N and also demonstrated that the ring of modular forms for the congruence subgroup 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 groupsEdit
A Fuchsian group Γ corresponds to the orbifold obtained from the quotient of the upper half-plane . By a stacky generalization of Riemann's existence theorem, 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 be the stabilizer orders of the stacky points of the stacky curve (equivalently, the cusps of the orbifold ) associated to Γ. If Γ has no nonzero odd weight modular forms, then the ring of modular forms is generated in weight at most and has relations generated in weight at most If Γ has a nonzero odd weight modular form, then the ring of modular forms is generated in weight at most and has relations generated in weight at most .
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).
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