Editing Modular form (section)

{{Short description|Analytic function on the upper half-plane with a certain behavior under the modular group}}
{{Redirect|Modular function|text=A distinct use of this term appears in relation to [[Haar measure#The modular function|Haar measure]]}}

{{Technical|date=February 2024}}
In [[mathematics]], a '''modular form''' is a [[holomorphic function]] on the [[Upper half-plane#Complex plane|complex upper half-plane]], <math>\mathcal{H}</math>, that roughly satisfies a [[functional equation]] with respect to the [[Group action (mathematics)|group action]] of the [[modular group]] and a growth condition. The theory of modular forms has origins in [[complex analysis]], with important connections with [[number theory]]. Modular forms also appear in other areas, such as [[algebraic topology]], [[sphere packing]], and [[string theory]].

Modular form theory is a special case of the more general theory of [[automorphic form]]s, which are functions defined on [[Lie group]]s that transform nicely with respect to the action of certain [[discrete subgroup]]s, generalizing the example of the modular group <math>\mathrm{SL}_2(\mathbb Z) \subset \mathrm{SL}_2(\mathbb R)</math>. Every modular form is attached to a [[Galois representation]].<ref name=":0">{{Cite news |last=Van Wyk |first=Gerhard |date=July 2023 |title=Elliptic Curves Yield Their Secrets in a New Number System |work=Quanta |url=https://www.quantamagazine.org/elliptic-curves-yield-their-secrets-in-a-new-number-system-20230706/?mc_cid=e612def96e&mc_eid=506130a407}}</ref>

The term "modular form", as a systematic description, is usually attributed to [[Erich Hecke]]. The importance of modular forms across multiple field of mathematics has been humorously represented in a possibly apocryphal quote attributed to [[Martin Eichler]] describing modular forms as being the fifth fundamental operation in mathematics, after addition, subtraction, multiplication and division.<ref>{{Cite web |last=Cepelewicz |first=Jordana |date=2023-09-21 |title=Behold Modular Forms, the 'Fifth Fundamental Operation' of Math |url=https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/ |access-date=2025-02-25 |website=Quanta Magazine |language=en}}</ref>