Editing Modular form (section)

==Rings of modular forms==
{{Main|Ring of modular forms}}
For a subgroup {{math|Γ}} of the {{math|SL(2, '''Z''')}}, the ring of modular forms is the [[graded ring]] generated by the modular forms of {{math|Γ}}. In other words, if {{math|M<sub>k</sub>(Γ)}} is the vector space of modular forms of weight {{mvar|k}}, then the ring of modular forms of {{math|Γ}} is the graded ring <math>M(\Gamma) = \bigoplus_{k > 0} M_k(\Gamma)</math>.

Rings of modular forms of congruence subgroups of {{math|SL(2, '''Z''')}} are finitely generated due to a result of [[Pierre Deligne]] and [[Michael Rapoport]]. Such rings of modular forms are generated in weight at most 6 and the relations are generated in weight at most 12 when the congruence subgroup has nonzero odd weight modular forms, and the corresponding bounds are 5 and 10 when there are no nonzero odd weight modular forms.

More generally, there are formulas for bounds on the weights of generators of the ring of modular forms and its relations for arbitrary [[Fuchsian group]]s.