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.

DefinitionEdit

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  .[1]

ExampleEdit

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.[1]

PropertiesEdit

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 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)Edit

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 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.[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]

ApplicationsEdit

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]

ReferencesEdit

  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.