Ring of modular forms

In mathematics, the '''ring of modular forms''' associated to a subgroup {{math|Γ}} of the special linear group {{math|SL(2, '''Z''')}} is the graded ring generated by the modular forms of {{math|Γ}}. The study of rings of modular forms describes the algebraic structure of the space of modular forms.

==Definition== Let {{math|Γ}} be a subgroup of {{math|SL(2, '''Z''')}} that is of finite index and let {{math|Mk(Γ)}} be the vector space of modular forms of weight {{mvar|k}}. The ring of modular forms of {{math|Γ}} is the graded ring <math display="inline">M(\Gamma) = \bigoplus_{k \geq 0} M_k(\Gamma)</math>.<ref name="Zagier"/>

==Example== The ring of modular forms of the full modular group {{math|SL(2, '''Z''')}} is freely generated by the Eisenstein series {{math|E4}} and {{math|E6}}. In other words, {{math|Mk(Γ)}} is isomorphic as a <math>\mathbb{C}</math>-algebra to <math>\mathbb{C}[E_4, E_6]</math>, which is the polynomial ring of two variables over the complex numbers.<ref name="Zagier"/>

==Properties== The ring of modular forms is a graded Lie algebra since the Lie bracket <math>[f,g] = kfg' - \ell f'g</math> of modular forms {{mvar|f}} and {{mvar|g}} of respective weights {{mvar|k}} and {{mvar|ℓ}} is a modular form of weight {{math|''k'' + ''ℓ'' + 2}}.<ref name="Zagier">{{cite book|last=Zagier |first=Don |author-link=Don Zagier |chapter=Elliptic Modular Forms and Their Applications |title=The 1-2-3 of Modular Forms |editor1-last=Bruinier |editor1-first=Jan Hendrik |editor1-link=Jan Hendrik Bruinier |editor2-last=van der Geer |editor2-first=Gerard |editor3-last=Harder |editor3-first=Günter |editor3-link=Günter Harder |editor4-last=Zagier |editor4-first=Don |publisher=Springer-Verlag |series=Universitext |pages=1–103 |isbn=978-3-540-74119-0 |doi=10.1007/978-3-540-74119-0_1 |chapter-url=https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf|year=2008 }}</ref> A bracket can be defined for the {{mvar|n}}-th derivative of modular forms and such a bracket is called a Rankin–Cohen bracket.<ref name="Zagier"/>

===Congruence subgroups of SL(2, Z)=== In 1973, Pierre Deligne and Michael Rapoport showed that the ring of modular forms {{math|M(Γ)}} is finitely generated when {{math|Γ}} is a congruence subgroup of {{math|SL(2, '''Z''')}}.<ref>{{cite book|last1=Deligne |first1=Pierre |author-link1=Pierre Deligne |last2=Rapoport |first2=Michael |author-link2=Michael Rapoport |chapter=Les schémas de modules de courbes elliptiques |title=Modular functions of one variable, II |publisher=Springer |orig-year=1973 |pages=143–316 |series=Lecture Notes in Mathematics |volume=349 |chapter-url=https://books.google.com/books?id=_L9sCQAAQBAJpg |isbn=9783540378556 |year=2009}}</ref>

In 2003, Lev Borisov and Paul Gunnells showed that the ring of modular forms {{math|M(Γ)}} is generated in weight at most 3 when <math>\Gamma</math> is the congruence subgroup <math>\Gamma_1(N)</math> of prime level {{mvar|N}} in {{math|SL(2, '''Z''')}} using the theory of toric modular forms.<ref>{{cite journal |last1=Borisov |first1=Lev A. |last2=Gunnells |first2=Paul E. |title=Toric modular forms of higher weight |journal=J. Reine Angew. Math. |volume=560 |year=2003 |pages=43–64 |arxiv=math/0203242|bibcode=2002math......3242B }}</ref> In 2014, Nadim Rustom extended the result of Borisov and Gunnells for <math>\Gamma_1(N)</math> to all levels {{mvar|N}} and also demonstrated that the ring of modular forms for the congruence subgroup <math>\Gamma_0(N)</math> is generated in weight at most 6 for some levels {{mvar|N}}.<ref>{{cite journal |last=Rustom |first=Nadim |title=Generators of graded rings of modular forms |journal=Journal of Number Theory |volume=138 |year=2014 |pages=97–118 |arxiv=1209.3864 |doi=10.1016/j.jnt.2013.12.008|s2cid=119317127 }}</ref>

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 {{math|Γ}} of {{math|SL(2, '''Z''')}} is generated in weight at most 6 with relations generated in weight at most 12.<ref name="vzb">{{cite book |title=The canonical ring of a stacky curve |last1=Voight |first1=John |last2=Zureick-Brown |first2=David |series=Memoirs of the American Mathematical Society |arxiv=1501.04657|year=2015 |bibcode=2015arXiv150104657V }}</ref> 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 {{math|Γ}} has some nonzero odd weight modular form.<ref name="lrz">{{cite journal |last1=Landesman |first1=Aaron |last2=Ruhm |first2=Peter |last3=Zhang |first3=Robin |title=Spin canonical rings of log stacky curves |journal = Annales de l'Institut Fourier |volume=66 |issue=6 |pages=2339–2383 |arxiv=1507.02643 |doi=10.5802/aif.3065|year=2016 |s2cid=119326707 }}</ref>

===General Fuchsian groups=== A Fuchsian group {{math|Γ}} corresponds to the orbifold obtained from the quotient <math>\Gamma \backslash \mathbb{H}</math> of the upper half-plane <math>\mathbb{H}</math>. By a stacky generalization of Riemann's existence theorem, there is a correspondence between the ring of modular forms of {{math|Γ}} and a particular section ring closely related to the canonical ring of a stacky curve.<ref name="vzb"/>

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 <math>e_i</math> be the stabilizer orders of the stacky points of the stacky curve (equivalently, the cusps of the orbifold <math>\Gamma \backslash \mathbb{H}</math>) associated to {{math|Γ}}. If {{math|Γ}} has no nonzero odd weight modular forms, then the ring of modular forms is generated in weight at most <math>6 \max(1, e_1, e_2, \ldots, e_r)</math> and has relations generated in weight at most <math>12 \max(1, e_1, e_2, \ldots, e_r)</math>.<ref name="vzb"/> If {{math|Γ}} has a nonzero odd weight modular form, then the ring of modular forms is generated in weight at most <math>\max(5, e_1, e_2, \ldots, e_r)</math> and has relations generated in weight at most <math>2\max(5, e_1, e_2, \ldots, e_r)</math>.<ref name="lrz"/>

==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.<ref name="bt">{{cite journal |title=Permutations of massive vacua |last1=Bourget |first1=Antoine |last2=Troost |first2=Jan |journal=Journal of High Energy Physics |volume=2017 |issue=42 |pages=42 |year=2017 |issn=1029-8479 |doi=10.1007/JHEP05(2017)042 |url=https://link.springer.com/content/pdf/10.1007%2FJHEP05%282017%29042.pdf|bibcode=2017JHEP...05..042B |arxiv=1702.02102 |s2cid=119225134 }}</ref> The stabilizers of superpotentials in N = 4 supersymmetric Yang–Mills theory are rings of modular forms of the congruence subgroup {{math|Γ(2)}} of {{math|SL(2, '''Z''')}}.<ref name="bt"/><ref>{{cite journal |last=Ritz |first=Adam |title=Central charges, S-duality and massive vacua of N = 1* super Yang-Mills |journal=Physics Letters B |year=2006 |volume=641 |issue=3–4 |pages=338–341 |doi=10.1016/j.physletb.2006.08.066 |arxiv=hep-th/0606050|s2cid=13895731 }}</ref>

==References== {{reflist}}

{{DEFAULTSORT:Ring of modular forms}} Category:Lie algebras Category:Modular forms Category:Number theory