# 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.

## Definition

Let Γ be a subgroup of SL(2, **Z**) that is of finite index and let M_{k}(Γ) be the ring of modular forms of weight k. The ring of modular forms of Γ is the graded ring .^{[1]}

## Example

The ring of modular forms of the full modular group SL(2, **Z**) is freely generated by the Eisenstein series E_{4} and E_{6}. In other words, M_{k}(Γ) is isomorphic as a
C
{\displaystyle \mathbb {C} }
-algebra , which is a polynomial ring of two variables.^{[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]}

### 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**).^{[2]}

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.^{[3]} 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.^{[4]}

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.^{[5]}

### General Fuchsian groups

A Fuchsian group Γ corresponds to the orbifold obtained from the quotient of the upper half-plane . By a generalization of Serre's GAGA due to John 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.^{[5]}^{[6]}

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 .^{[5]}

## 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.^{[7]} 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**).^{[7]}^{[8]}

## References

- ^
^{a}^{b}^{c}^{d}Zagier, Don (2008). "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; Rapoport, Michael (1973). "Les schémas de modules de courbes elliptiques".*Modular functions of one variable, II*. Lecture Notes in Mathematics.**349**. Berlin: Springer-Verlag. pp. 143–316.**^**Borisov, Lev A.; Gunnells, Paul E. (2003). "Toric modular forms of higher weight".*J. Reine Angew. Math.***560**: 43–64. arXiv:math/0203242. Bibcode:2002math......3242B.**^**Rustom, Nadim (2014). "Generators of graded rings of modular forms".*Journal of Number Theory*.**138**: 97–118. arXiv:1209.3864. doi:10.1016/j.jnt.2013.12.008.- ^
^{a}^{b}^{c}Landesman, Aaron; Ruhm, Peter; Zhang, Robin (2016). "Spin canonical rings of log stacky curves".*Annales de l'Institut Fourier*.**66**(6): 2339–2383. arXiv:1507.02643. doi:10.5802/aif.3065. **^**Voight, John; Zureick-Brown, David (2015).*The canonical ring of a stacky curve*. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V.- ^
^{a}^{b}Bourget, Antoine; Troost, Jan (2017). "Permutations of massive vacua" (PDF).*Journal of High Energy Physics*.**2017**(42): 42. arXiv:1702.02102. Bibcode:2017JHEP...05..042B. doi:10.1007/JHEP05(2017)042. ISSN 1029-8479. **^**Ritz, Adam (2006). "Central charges, S-duality and massive vacua of N = 1* super Yang-Mills".*Physics Letters B*.**641**(3–4): 338–341. arXiv:hep-th/0606050. doi:10.1016/j.physletb.2006.08.066.