# 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 Mk(Γ) be the ring of modular forms of weight k. The ring of modular forms of Γ is the graded ring ${\displaystyle M(\Gamma )=\bigoplus _{k>0}M_{k}(\Gamma )}$ .[1]

## 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 ${\displaystyle \mathbb {C} }$ -algebra ${\displaystyle \mathbb {C} [E_{4},E_{6}]}$ , which is a polynomial ring of two variables.[1]

## Properties

The ring of modular forms is a graded Lie algebra since the Lie bracket ${\displaystyle [f,g]=kfg'-\ell f'g}$  of modular forms f and g of respective weights k and is a modular form of weight k + ℓ + 2.[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 ${\displaystyle \Gamma }$  is the congruence subgroup ${\displaystyle \Gamma _{1}(N)}$  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 ${\displaystyle \Gamma _{1}(N)}$  to all levels N and also demonstrated that the ring of modular forms for the congruence subgroup ${\displaystyle \Gamma _{0}(N)}$  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 ${\displaystyle \Gamma \backslash \mathbb {H} }$  of the upper half-plane ${\displaystyle \mathbb {H} }$ . 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.[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 ${\displaystyle e_{i}}$  be the stabilizer orders of the stacky points of the stacky curve (equivalently, the cusps of the orbifold ${\displaystyle \Gamma \backslash \mathbb {H} }$ ) associated to Γ. If Γ has no nonzero odd weight modular forms, then the ring of modular forms is generated in weight at most ${\displaystyle 6\max(3,e_{1},e_{2},\ldots ,e_{r})}$  and has relations generated in weight at most ${\displaystyle 12\max(3,e_{1},e_{2},\ldots ,e_{r})}$  If Γ has a nonzero odd weight modular form, then the ring of modular forms is generated in weight at most ${\displaystyle \max(5,e_{1},e_{2},\ldots ,e_{r})}$  and has relations generated in weight at most ${\displaystyle 2\max(5,e_{1},e_{2},\ldots ,e_{r})}$ .[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

1. ^ 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.
2. ^ 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.
3. ^ 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.
4. ^ 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.
5. ^ 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.
6. ^ Voight, John; Zureick-Brown, David (2015). The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V.
7. ^ 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.
8. ^ 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.