Ring of modular forms explained

In mathematics, the ring of modular forms associated to a subgroup of the special linear group is the graded ring generated by the modular forms of . The study of rings of modular forms describes the algebraic structure of the space of modular forms.

Definition

Let be a subgroup of that is of finite index and let be the vector space of modular forms of weight . The ring of modular forms of is the graded ring $M(\backslash Gamma)\; =\; \backslash bigoplus\_\; M\_k(\backslash Gamma)$.

Example

The ring of modular forms of the full modular group is freely generated by the Eisenstein series and . In other words, is isomorphic as a

C -algebra to

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] = kfg' - \ell f'g of modular forms and of respective weights and is a modular form of weight .^[1] A bracket can be defined for the -th derivative of modular forms and such a bracket is called a Rankin–Cohen bracket.^[1]

Congruence subgroups of SL(2, Z)

In 1973, Pierre Deligne and Michael Rapoport showed that the ring of modular forms is finitely generated when is a congruence subgroup of .^[2]

In 2003, Lev Borisov and Paul Gunnells showed that the ring of modular forms is generated in weight at most 3 when

\Gamma is the congruence subgroup \Gamma _{1(N)} of prime level in using the theory of toric modular forms.^[3] In 2014, Nadim Rustom extended the result of Borisov and Gunnells for \Gamma _{1(N)} to all levels 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 .^[4]

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 of is generated in weight at most 6 with relations generated in weight at most 12.^[5] 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 has some nonzero odd weight modular form.^[6]

General Fuchsian groups

A Fuchsian group corresponds to the orbifold obtained from the quotient

\Gamma \backslash H of the upper half-plane H . By a stacky generalization of Riemann's existence theorem, there is a correspondence between the ring of modular forms of and the a particular section ring closely related to the canonical ring of a stacky curve.^[5]

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 H ) associated to . If has no nonzero odd weight modular forms, then the ring of modular forms is generated in weight at most 6 max ( 1 , e _{1,} e _{2,} \ldots, e _{r)} and has relations generated in weight at most 12 max ( 1 , e _{1,} e _{2,} \ldots, e _{r)} .^[5] If 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 2max ( 5 , e _{1,} e _{2,} \ldots, e _{r)} .^[6]

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 of .^{[7] [8]}

Notes and References

1. Book: Zagier, Don . Don Zagier . Elliptic Modular Forms and Their Applications . The 1-2-3 of Modular Forms . Bruinier . Jan Hendrik . Jan Hendrik Bruinier . van der Geer . Gerard . Harder . Günter . Günter Harder . Zagier . Don . Springer-Verlag . Universitext . 1–103 . 978-3-540-74119-0 . 10.1007/978-3-540-74119-0_1 . https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf. 2008 .
2. Book: Deligne . Pierre . Pierre Deligne . Rapoport . Michael . Michael Rapoport . Les schémas de modules de courbes elliptiques . Modular functions of one variable, II . Springer . 1973 . 143–316 . Lecture Notes in Mathematics . 349 . https://books.google.com/books?id=_L9sCQAAQBAJpg . 9783540378556 . 2009.
3. Borisov . Lev A. . Gunnells . Paul E. . Toric modular forms of higher weight . . 560 . 2003 . 43–64 . math/0203242. 2002math......3242B .
4. Rustom . Nadim . Generators of graded rings of modular forms . Journal of Number Theory . 138 . 2014 . 97–118 . 1209.3864 . 10.1016/j.jnt.2013.12.008. 119317127 .
5. Book: The canonical ring of a stacky curve . Voight . John . Zureick-Brown . David . . 1501.04657. 2015 . 2015arXiv150104657V .
6. Landesman . Aaron . Ruhm . Peter . Zhang . Robin . Spin canonical rings of log stacky curves . . 66 . 6 . 2339–2383 . 1507.02643 . 10.5802/aif.3065. 2016 . 119326707 .
7. Permutations of massive vacua . Bourget . Antoine . Troost . Jan . Journal of High Energy Physics . 2017 . 42 . 42 . 2017 . 1029-8479 . 10.1007/JHEP05(2017)042 . 2017JHEP...05..042B . 1702.02102 . 119225134 .
8. Ritz . Adam . Central charges, S-duality and massive vacua of N = 1* super Yang-Mills . Physics Letters B . 2006 . 641 . 3–4 . 338–341 . 10.1016/j.physletb.2006.08.066 . hep-th/0606050. 13895731 .