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 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 Γ of SL(2, Z) 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 Γ has some nonzero odd weight modular form.^[6]

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]

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 ${\displaystyle 6\max(1,e_{1},e_{2},\ldots ,e_{r})}$ and has relations generated in weight at most ${\displaystyle 12\max(1,e_{1},e_{2},\ldots ,e_{r})}$.^[5] 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 ${\displaystyle 2\max(5,e_{1},e_{2},\ldots ,e_{r})}$.^[6]