Ring of modular forms

Let {{math|Γ}} be a subgroup of {{math|SL(2, Z)}} that is of finite index and let {{math|M_k(Γ)}} be the vector space of modular forms of weight {{mvar|k}}. The ring of modular forms of {{math|Γ}} is the graded ring $M(\backslash Gamma)\; =\; \backslash bigoplus\_\{k\; \backslash geq\; 0\}\; M\_k(\backslash Gamma)$.

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|M_k(Γ)}} is isomorphic as a $\backslash mathbb\{C\}$-algebra to $\backslash mathbb\{C\}[E\_4,\; E\_6]$, which is the polynomial ring of two variables over the complex numbers.

The ring of modular forms is a graded Lie algebra since the Lie bracket $[f,g]\; =\; kfg\text{'}\; -\; \backslash ell\; f\text{'}g$ 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}}.{{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 }} A bracket can be defined for the {{mvar|n}}-th derivative of modular forms and such a bracket is called a Rankin–Cohen bracket.

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)}}.{{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}}

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 $\backslash Gamma$ is the congruence subgroup $\backslash Gamma\_1(N)$ of prime level {{mvar|N}} in {{math|SL(2, Z)}} using the theory of toric modular forms.{{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 }} In 2014, Nadim Rustom extended the result of Borisov and Gunnells for $\backslash Gamma\_1(N)$ to all levels {{mvar|N}} and also demonstrated that the ring of modular forms for the congruence subgroup $\backslash Gamma\_0(N)$ is generated in weight at most 6 for some levels {{mvar|N}}.{{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 }}

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.{{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 }} 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.{{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 }}

A Fuchsian group {{math|Γ}} corresponds to the orbifold obtained from the quotient $\backslash Gamma\; \backslash backslash\; \backslash mathbb\{H\}$ of the upper half-plane $\backslash mathbb\{H\}$. 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.

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 $\backslash Gamma\; \backslash backslash\; \backslash mathbb\{H\}$) 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 $6\; \backslash max(1,\; e\_1,\; e\_2,\; \backslash ldots,\; e\_r)$ and has relations generated in weight at most $12\; \backslash max(1,\; e\_1,\; e\_2,\; \backslash ldots,\; e\_r)$. If {{math|Γ}} has a nonzero odd weight modular form, then the ring of modular forms is generated in weight at most $\backslash max(5,\; e\_1,\; e\_2,\; \backslash ldots,\; e\_r)$ and has relations generated in weight at most $2\backslash max(5,\; e\_1,\; e\_2,\; \backslash ldots,\; e\_r)$.

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.{{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 }} 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)}}.{{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 }}

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Category:Lie algebras

Category:Modular forms

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